Numerical investigation on vane leading edge impingement cooling performance under influences of ribs established between adjacent jets

Abstract

Numerical studies on the vane leading edge jet impingement cooling are conducted to analyzed the influence of the cylindrical ribs applied between adjacent jets. A one-way inlet coolant supply channel and a one-way exit semicircular target channel are connected through single array of 10 jet holes. The gradually accumulated cross flow greatly decreases the impingement heat transfer intensity at the target channel downstream region. Therefore, the influences of rib diameters (D_r/D = 0.25, 0.5, 0.75, 1.0) and three rib to jet distances (P/D = 1.2, 1.5, 1.8) are numerically studied to decrease the effect of the cross flow and increase the impingement heat transfer performance as well as the heat transfer uniformity. The Reynolds-averaged Navier-Stokes equations (RANS) method coupled with Realizable k–ε turbulence model are used to complete the numerical simulation, the numerical method is carefully validated. The pressure and velocity distributions are illustrated. The effects of ribs on the flow characteristics inside the semicircular target channel are discussed and three major vortical structures are captured and compared. The two heat transfer enhancement mechanisms which depend on the rib to jet distance are also analyzed in detail. Results indicate that, with the application of ribs, the area-averaged Nusselt number increases about 4.3%–11.3% and the impingement heat transfer is more uniformly distributed along the target channel axis. When pursuing a larger heat transfer enhancement and a larger performance evaluation criterion (PEC), a larger rib diameter coupled with a larger rib to downstream jet distance is preferred, the largest PEC enhancement is about 8.1%.

Keywords

Impingement cooling leading edge cross flow cylindrical rib numerical simulation

Introduction

For state-of-the-art engines, gas temperature at the turbine inlet has increased to more than 2000 K. To ensure the engine performance and the engine operating life, cooling strategies are commonly applied on the high temperature components. The vane leading edge suffers the direct impingement of hot gas, bears the highest heat flux and most complex flow field. Therefore, efficient and reliable thermal protection methods should be taken at the vane leading edge. In all the existing cooling methods, impingement cooling method can rebuild the boundary layer at the impingement region and enhance the heat transfer flux between coolants and target surface, hence the impingement cooling method is well-suited for the vane leading edge thermal protection.

Due to the complex flow features of the impingement cooling, the jet development process as well as the corresponding cooling performance is determined by a lot of influencing factors,^1–3 such as the interference of adjacent jets, the cross flow scheme and its effect, the shape of both the jet hole and the target channel and the like. Forster and Weigand⁴ explored the cooling performance of single jet array based on a curved target surface. They found that the cooling performance was decreased but the local heat transfer distribution was homogenized by the cross flow. Tepe⁵ numerically investigated the impingement cooling performance of extended jet holes. Results showed that the extended jet hole structure is beneficial to the cooling performance on a semicircular target surface, a smaller jet to target distance was more preferred. Wright et al.⁶ conducted an investigation on impingement cooling with varied jet positions. Pachpute and Premachandran⁷ explored the impingement cooling under different jet Reynolds numbers, jet impingement distances and diameter ratios of target channel to jet hole. Heat transfer correlations were then presented accordingly. Tepe et al.^5,8 came up with an extended jet structure and found the impingement cooling performance under the staggered jet arrangement is increased due to the decreased jet to target distance.

Limited by the coolant supply mode and the narrow space at the leading edge, there is only one outlet of the target channel, the cross flow is therefore concentrated in one direction. Since the cooling jet will be impacted and squeezed by the cross flow, the impingement cooling performance is significantly reduced. Lee et al.⁹ indicated that the target surface heat transfer is reduced by the gradually accumulated cross flow under the multiple jet arrays configuration. Besides, on account of the enhanced interference between contiguous impinging jets and the locally enhanced turbulent transport, the local Nusselt number was firstly increased by the cross flow. Yang et al.¹⁰ indicated that, affected by the vortical structures in both the axial and spanwise directions, the inverse temperature gradient is generated which results in the temperature nonuniformity. Besides, a larger jet Reynolds number and a shorter jet impingement distance yielded a better temperature uniformity. Otero-Pérez et al.¹¹ conducted a comprehensive investigation on multiple jet arrays cooling performance with large eddy simulations. The cross flow caused a jet deflection and reduced the Nusselt number peak value, the cooling performance in the areas less affected by the jet impingement was enhanced by the cross flow. Besides, adding turbulent fluctuations at the coolant inlet increased the effect of cross flow. Chen et al.¹² compared the impingement cooling performance among several cross flow schemes. With the application of film holes, more coolants were ejected out when the cross flow is developed along single direction, the cooling performance was then enhanced.

As the uneven distributed heat transfer performance caused by the one-way developed cross flow was undesirable,¹³ many researchers conducted investigations to decrease the negative influences of the cross flow. Terzis et al.¹⁴ found the negative effect of the cross flow could be adjusted by the jet hole diameter along the streamwise direction. An initially enlarged and then reduced jet hole diameter arrangement was also proposed by Ji et al.¹⁵ Zhou et al.¹⁶ came up with a jet array configuration with varied jet diameter based on three cross flow development stages to counter the negative effect of cross flow. The relation between the cross flow development and the jet diameter distribution was established. The non-uniformity coefficient among all impingement regions was successfully decreased from 9%–12% to 3%–4%. Ligrani et al.¹⁷ tested the impingement cooling performance under different initial blowing ratios and contraction ratios. They proposed three competing phenomena that affect the target surface heat transfer. Chi et al.¹⁸ came up with a novel impingement plate structure to direct the cross flow into the space between adjacent jets. As such, the impact intensity of the cross flow on the jet flow was impaired, the target surface heat transfer was also more uniformly distributed with optimum coolant consumption.

For the past few years, rib structures were also applied to diminish the negative effects of the cross flow and increase both the cooling performance and the target surface heat transfer uniformity. He et al.¹⁹ applied several kinds of cross flow diverters to obstruct the cross flow development with the cost of increased friction losses. The influences of the diverter location and dimension were also studied. Salem et al.^20,21 applied cylindrical, rectangular and ribbed diverters to remove the unfavorable effects of the cross flow. The Nusselt number was found to increase about 6.6%–12%, but the friction factor was increased about 9.9%–24.3% at the same time. He et al.²² applied a corrugated target surface for a better impingement cooling performance. Madhavan et al.²³ used U-shaped ribs to protect the cooling jets from being impacted by the cross flow. There was about 9%–15% enhancement in heat transfer performance under a fixed pumping power. Ren et al.²⁴ experimentally investigated the effects of toughened target surface and found the near wall mixing is enhanced and the turbulence intensity is increased due to the reconstructed viscous sublayer. Yalçınkaya and Durmaz²⁵ performed numerical studies to analyze the effect of the extended jet nozzle on the cooling performance on a rib-roughened curved target surface. Rectangular cross-sectional V-shaped ribs were applied. The effects of rib height and rib angle were also studied. Based on the numerical results, the heat transfer performance as well as the heat transfer uniformity was increased. The highest thermal performance criterion was obtained under the rib height of H_r/d = 0.2 and the jet nozzle extend distance of G/d = 2.0. Kim et al.²⁶ experimentally and numerically compared the negative effect of the cross flow of the castellated jet impingement with and without rib structures on the target surface. The influence of jet spacing and jet impingement distance were also studied. Results showed that, due to the increased target channel cross-sectional area, the pressure drop inside the target channel was decreased. However, the rib structures on the target surface yielded little effects on the heat transfer performance. Fu et al.²⁷ applied W-shaped micro-ribs to reduce the effect of the cross flow and enhance the impingement cooling performance. The results showed that the ribs can separate the cross flow and reduce the effect of the cross flow, the longitudinal vortex generated by the cross flow could further enhance the convective heat transfer. The highest heat transfer enhancement was about 39.7%–48.9%. Fathi and Nejat²⁸ proposed an impingement cooling configuration in which two rows of ribs were established between adjacent impingement regions. The conjugate cooling performance was sensitive to the rib location and when the ribs were perpendicular to the cross flow, the overall cooling effectiveness was further increased. Shukla et al.²⁹ investigated the flow and heat transfer characteristics of impingement cooling on a surface covered with detached ribs based on the large eddy simulation method. Results showed that, the heat transfer was enhanced near the clearance between the ribs and the target surface due to the increased velocity and turbulence.

Most of the previous studies put the ribs in front of the impinging jets to decrease the influence of the cross flow. However, by applying the rib structure, the flow features inside the target channel become more complex and the flow losses are also enhanced. Therefore, it is important to conduct a comprehensive investigation on the influencing mechanism of ribs under different rib to jet distances and rib dimensions. Thus, this paper aims to find out a more rational layout of the rib structure to gain a better impingement cooling performance and a more uniformly distributed heat transfer on the target surface. To achieve this goal, numerical studies are conducted inside a simplified leading edge target channel under three jet to rib distances and four rib diameters. The impingement cooling model without applying ribs is used as a benchmark case for the convenience of the comparison. The major vortical structures coupled with pressure and velocity distributions are illustrated to reflect the cross flow development and the effects of ribs. The heat transfer enhancement mechanisms by applying ribs between adjacent jets are then analyzed accordingly.

Numerical method

Simplified leading edge impingement cooling model

The simplified geometry of the leading edge impingement cooling is shown in Figure 1. A half domain numerical structure is applied to reduce the grid quantity. The cooling air impinge into the semicircular target channel successively through 10 jet holes, and the jet diameter is D = 10 mm. The length of impinging hole L, the axial distance between adjacent jets S_x and the jet impinging distance Z are 1D, 3D and 3D, respectively. The target channel radius is 3D. An effective cooling region between the target surface centerline and 75° line is defined to more reasonably reflect the leading edge impingement cooling performance. To better show the computational domain used in this study, a 3D view is also provided as a supplement.

Figure 1.

Simplified impingement cooling geometry at vane leading edge.

Cylindrical ribs are added in front of each jet hole (except the first jet hole as there is no upstream cross flow) to decrease the influence of upstream cross flow on downstream cooling jet. As shown in Table 1, the influences of the jet to rib distance P (from the centerline of a jet hole to the centerline of an adjacent upstream rib) and the rib diameter D_r are investigated. Since a too small jet to rib distance or a too large rib diameter will excessively affect both the jet development process and the boundary layer reconstruction process after jet impingement, only three jet to rib distances (1.2D, 1.5D, 1.8D) and four rib diameters (0.25D, 0.50D, 0.75D, 1.0D) are applied in this paper. Besides, to facilitate the comparison among different cases, the impingement cooling geometry without applying the ribs (D_r = 0) is selected as a benchmark case.

Table 1.

Brief summary of geometric parameter settings.

Geometric parameter	P	D _r	S_x	Z	L
D = 10 mm	1.2D, 1.5D, 1.8D	0.25D, 0.5D, 0.75D, 1.0D	3D	3D	1D

Governing equations

The steady states governing equations that describe the incompressible flow inside the target channel are written as follow

Continuity equation

\frac{\partial u_{j}}{\partial x_{j}} = 0

(1)

Momentum equation

ρ u_{j} \frac{\partial u_{i}}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} (τ_{ij} - ρ \bar{{u'}_{i} {u'}_{j}}) + S_{M}

(2)

Energy equation

\begin{matrix} ρ u_{j} \frac{\partial h_{tot}}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} (λ \frac{\partial T}{\partial x_{j}} - ρ \bar{{u'}_{j} h}) + \frac{\partial}{\partial x_{j}} [u_{i} (τ_{ij} - ρ \bar{{u_{i}}^{'} {u'}_{j}})] + S_{E} \end{matrix}

(3)

where u_i and u_j are the time mean velocity, $u'_{i}$ and $u'_{j}$ are the time fluctuating velocity, p and τ are the normal stress and the stress tensor respectively, h_tot is the total enthalpy, S_M and S_E are source terms in momentum and energy equations respectively. The heat flux from the target surface into the coolant is considered in S_E term and there is no source term in momentum equation. Air ideal gas is selected as the coolant. Piecewise polynomial fitting is used to correct the temperature dependent specific heat. Kinetic theory is used to correct the temperature dependent thermal conductivity. Sutherland’s law is used to correct the temperature dependent viscosity.

Numerical boundary condition settings

The numerical boundary conditions as listed in Table 2 are based on Yang et al.’s^30,31 experimental work. The major dimensions (D, L, S_x, Z) of Yang’s experimental model are all same with that of the numerical model studied in this work. The average jet Reynolds number of 10 jet holes (with the characteristic length of D) is 15,000 and the relevant total mass flow rate at the inlet is 1.225 × 10⁻² kg·s⁻¹ with a turbulence intensity of 1%. The coolant temperature at inlet is T_c,in =348.15 K, the temperature on all target channel surfaces is fixed as T_w = 419.15 K, the surfaces of jet holes and coolant supply channel are adiabatic. The constant static pressure condition is applied at the target channel outlet with P_out = 0.1 MPa.

Table 2.

Boundary condition settings.

$\overset{\cdot}{m}$ (kg·m⁻³)	Re _D	T_c,_in (K)	T _w (K)	P _out (MPa)	Tu _in
1.225 × 10⁻²	15,000	348.15	419.15	0.1	1%

Grid independence analysis

The hexahedral structured grid is used to segment the fluid domain for all impingement cooling structures with ICEM 2023. Figure 2 shows the meshing details under p = 1.2D and D_r = 0.5D. The boundary layers are applied on all wall surfaces with a stretching ratio of 1.1, y⁺ < 1.0 is guaranteed and the layer number is about 24. To enhance the computational accuracy, outer O-blocks as well as the near edge mesh refinement is applied around the cylindrical ribs and jet holes. To ensure a convergent computing result, the calculation ends when the residuals of both the continuity and velocity equations fall below 10⁻⁴. SIMPLEC scheme is used for the pressure-velocity coupling, second order upwind scheme is applied for the spatial discretization.

Figure 2.

Meshing details.

Grid independence studies are then conducted based on the calculated Nusselt number along the centerline of the target surface. The Nusselt number is written as

N u_{D} = \frac{D \cdot q_{w}}{λ (T_{w} - T_{c, in})}

(4)

where q_w is the heat flux, λ is the coolant thermal conductivity under T_c,in. Figure 3(a) shows the predicted result under p = 1.2D and D_r = 0.5D. The Nu_D curve predicted by the 2.5 million grids is significantly different from those predicted by other grids, especially at the impingement region and at the downstream region of ribs. With the increase of the grids number, the predicted Nu_D curve gradually converges. The Nu_D at impingement regions predicted by the 5.0 and 7.5 million grids differs about 5%, and the difference is less than 0.2% between 7.5 and 10.0 million grids. Therefore, the 7.5 million grids are selected to conduct the subsequent numerical investigations. About 7.5 million grids are also selected for other cases according to the predicted centerline Nu_D distribution which is not shown any more. GCI error³² of the selected grids is then analyzed to further ensure the calculation accuracy, GCI error is defined as

GC I_{12} = \frac{F_{s} | ε |}{r^{p} - 1}

(5)

where the safety factor F_s and the grid refinement ratio r are 1.25 and 2, respectively. ε is the relative error which is written as

ε = \frac{f_{2} - f_{1}}{f_{1}}

(6)

where f₁ and f₂ are the predicted value under the relevant normalized grid spacings, the convergence order p is then calculated as

p = \ln (\frac{f_{3} - f_{2}}{f_{2} - f_{1}}) / \ln (r)

(7)

Figure 3.

Grids independence analysis: (a) predicted centerline Nu_D distribution and (b) predicted area-averaged Nu_D.

The Richardson extrapolation of the predicted result is then written as

f_{h = 0} ≅ f_{1} + \frac{f_{1} - f_{2}}{r^{p} - 1}

(8)

Figure 3(b) shows the area-averaged Nusselt number $\bar{\bar{Nu}}$ of the effective cooling region, the predicted result of the selected grids is f₁. Results indicate that GCI₁₂ is no larger than 1% under all D_r values (D_r = 0 represents the benchmark case without the ribs). GCI error is also analyzed under p = 1.5D and 1.8D, the predicted $\bar{\bar{Nu}}$ indicates the similar conclusions, so they are not shown here anymore.

Applicability analysis of turbulence model

Numerical solution is obtained by using a steady approach of Reynolds-averaged Navier-Stokes equations (RANS) method in commercial software Fluent 2023. Turbulence models are firstly validated by comparing predicted results and the experimental data from Yang et al.’s^30,31 work in which the same boundary conditions are applied.

Figure 4(a) shows the predicted spanwise-averaged Nusselt number ( $\bar{Nu}$ ) with different turbulence models under D_r = 0. Enhanced wall function is applied as the y⁺ < 1 is strictly ensured on all wall surfaces. As can be seen, k–ω equation based models overestimate ten $\bar{Nu}$ peak values, S–A and realizable k–ε models make better predictions but also overestimate the first two $\bar{Nu}$ peak values which may due to the predictive bias of the coolant mass flow rate distribution, and the biggest deviation is about 35%. Realizable k–ε models both yields a less than 20% deviation at 12 < x/D < 24 region, and the locations of 10 $\bar{Nu}$ peak values are also better predicted. The Florschuetz et al.’s³³ model is also applied with the selected geometric parameters of (5, 4, 3), that is, the streamwise jet pitch is 5D (the minimum value that can be selected, 3D of Yang’s geometry), the spanwise pitch is 4D (the minimum value that can be selected, 3.85D of Yang’s geometry), and the jet impinging distance is 3D (3D of Yang’s geometry). The $\bar{Nu}$ value obtained by Florschuetz’s model are generally at the same level with the predicted value of Realizable k–ε turbulence model. However, Florschuetz’s model suggests the $\bar{Nu}$ peak values tend to decrease monotonically along x/D direction which is different with the predicted values by numerical calculation. This may due to the difference of the jet pitch and the channel geometry between Florschuetz’s model and Yang’s geometry. Llucià et al.³⁴ also indicated that, when the cross flow starts to accumulate, the impinging jet is more disturbed, the development process of the impinging jet is accelerated which lead to an enhanced impingement heat transfer. However, this phenomenon is not captured by the Florschuetz’s model. Figure 4(b) indicates that the Realizable k–ε turbulence model can better predict the Nu_D contour on the target surface.

Figure 4.

Numerical model validation (concave target channel): (a) spanwise averaged Nu_D and (b) Nu_D contour.

The experimental work (5, 5, 3) by Fechter et al.³⁵ is used to further ensure the applicability of Realizable k–ε turbulence model. Figure 5(a) shows the Nu_D value along the centerline. SST k–ω turbulence model again overestimates the impingement heat transfer intensity, Realizable k–ε generally makes a better prediction. Figure 5(b) suggests that the Realizable k–ε turbulence model could better reproduce the Nu_D distribution. Consequently, Realizable k–ε turbulence model based on the RANS method is selected to conduct numerical studies on the impingement heat transfer performance under the effect of cylindrical ribs applied between adjacent jets.

Figure 5.

Numerical model validation (rectangular target channel)³⁵: (a) Nu_D distribution along centerline and (b) Nu_D contour.

Results and heat transfer enhancement mechanism analysis

Performance evaluation criterion comparison

The jet mass flow rate distribution is firstly determined since its great influences on the flow development process and the impingement intensity. The mass flux ratio G_c/G_j is selected to show the cross flow development process which is defined as

\frac{G_{c}}{G_{j}} = \frac{m_{c, i} / A}{m_{j, i} / A_{j}}

(9)

where m_c,_i is the cross flow mass flow rate in front of the ith jet, m_j,_i is the jet mass flow rate through the ith hole. A_j and A are the cross section area of the jet hole and the target channel, respectively. Figure 6 shows the G_c/G_j ratio under p = 1.2D and 1.8D. The results of p = 1.5D are not shown here anymore since the final conclusion does not change. G_c/G_j of first hole is 0 since the target channel has only one exit. With the continuous impingement of downstream jets, the cross flow gradually accumulates along target channel axis. Results show that all G_c/G_j curves are almost the same before the seventh jet. The largest deviation appears at the last jet, however, the relevant deviation of the jet mass flow rate is still less than 4.0% among all cases. Therefore, the coolant mass flow rate distribution through ten jet holes and the basic cross flow development process are not obviously affected by either the rib locations or diameters, the flow features and the impingement heat transfer with ribs established between adjacent jets can be fairly compared under different P and D_r.

Figure 6.

Comparison of the mass flux ratio: (a) p = 1.2D and (b) p = 1.8D.

Table 3 compares the total pressure drop ΔP_t between the inlet and outlet of the numerical model. Generally, a larger jet impingement intensity leads to a larger ΔP_t. In the benchmark case of D_r = 0.00D, ΔP_t is about 1014 Pa. With the increase of D_r, the interference between ribs and cross flow becomes stronger, the impinging jet is less affected by the cross flow, ΔP_t increases about 5.7%–9.1% from D_r = 0.00D to D_r = 1.00D. Under the same D_r, ΔP_t is not heavily affected by P when D_r < 0.75D. It suggests that the relative location between ribs and adjacent jets has small effects on the flow features inside the target channel under a small D_r. However, when D_r = 0.75D, ΔP_t increases a lot with the increase of P. When D_r = 1.00D, ΔP_t under p = 1.8D is much larger than that under p = 1.2D and 1.5D. It suggests that the main flow features are also heavily affected by the rib location under a larger D_r.

Table 3.

Total pressure drop under different P and D_r.

ΔP_t (Pa)	D _r = 0.00D	D _r = 0.25D	D _r = 0.50D	D _r = 0.75D	D _r = 1.00D
p = 1.2D		1020	1035	1047	1072
p = 1.5D	1014	1015	1040	1054	1077
p = 1.8D		1020	1029	1061	1106

Table 4 shows the area-averaged Nusselt number $\bar{\bar{Nu}}$ at the effective cooling region. A larger $\bar{\bar{Nu}}$ suggests a better impingement heat transfer performance. With the increase of D_r, the upstream cross flow is more obstructed by ribs, the adverse influence of the cross flow on the impinging jet is decreased, $\bar{\bar{Nu}}$ increases about 4.3%–11.3% from D_r = 0.00D to D_r = 1.00D. Under the same D_r, $\bar{\bar{Nu}}$ is not heavily affected by P when D_r < 0.75D. It suggests that the jet impingement intensity is not severely changed by P under a smaller D_r. However, when D_r = 0.75D, $\bar{\bar{Nu}}$ under p = 1.8D is much larger than that under p = 1.2D and 1.5D. When D_r = 1.00D, $\bar{\bar{Nu}}$ increases about 6.7% from p = 1.2D to p = 1.8D. It again suggests that the flow features and the jet impingement intensity are heavily affected by P under a larger D_r. Based on the numerical results, a correlation between the area-averaged Nusselt number $\bar{\bar{Nu}}$ and the rib parameter P and D_r is also established, the error between the numerical results and the correlation is less than 3%,

\bar{\bar{Nu}} = 45.9337 \cdot {(\frac{P}{D})}^{0.0543} \cdot {(\frac{D_{r}}{D})}^{0.0322}

(10)

Table 4.

Area-averaged Nusselt number.

$\bar{\bar{Nu}}$	D _r = 0.00D	D _r = 0.25D	D _r = 0.50D	D _r = 0.75D	D _r = 1.00D
p = 1.2D		45.19	45.61	45.69	45.94
p = 1.5D	44.06	45.06	45.54	45.77	47.11
p = 1.8D		45.10	45.36	47.18	49.04

The Performance Evaluation Criterion (PEC)⁸ is then caluculated to make an overall evaluation of different cases, PEC is defined as

\frac{\bar{\bar{Nu}}}{{\bar{\bar{Nu}}}_{0}} / \frac{Δ P_{t}}{Δ P_{t, 0}}

(11)

where ${\bar{\bar{Nu}}}_{0}$ and $Δ P_{t, 0}$ are the area-averaged Nusselt number and total pressure drop of the benchmark impingement cooling case without applying the ribs, respectively. As shown in Table 5, PEC is enhanced by applying the ribs under all D_r and P. Under p = 1.2D, with the increase of D_r, PEC increases a little bit from D_r = 0.25D to D_r = 0.50D. As the flow inside the target channel becomes more chaotic with the further increase of D_r, ΔP_t increases monotonically, thus the PEC decreases gradually when D_r > 0.50D. Under p = 1.5D, PEC tends to increase with the increase of D_r. Under p = 1.8D, the PEC increases a lot with the increase of D_r, especially when D_r > 0.50D. Under the same D_r, PEC is again not heavily affected by P when D_r < 0.75D. When D_r = 0.75D, PEC under p = 1.8D is much larger than that under p = 1.2D and 1.5D. When D_r = 1.00D, the PEC increases about 5.7% from p = 1.2D to p = 1.8D. It seems that, a larger rib diameter is preferred when the rib is located much closer to their upstream jets and a smaller rib diameter is preferred when the ribs are located much closer to their downstream jets. The best PEC occurs at D_r = 1.00D when p = 1.8D. It can be concluded that, when pursuing a smaller flow loss, a smaller rib diameter coupled with a smaller rib to downstream jet distance are preferred. When pursuing a larger heat transfer enhancement and a larger PEC, a larger rib diameter coupled with a larger rib to downstream jet distance are preferred.

Table 5.

Comparison of performance evaluation criterion (PEC).

$\frac{\bar{\bar{Nu}}}{{\bar{\bar{Nu}}}_{0}} / \frac{Δ P_{t}}{Δ P_{t, 0}}$	D _r = 0.00D	D _r = 0.25D	D _r = 0.50D	D _r = 0.75D	D _r = 1.00D
p = 1.2D		1.024	1.028	1.026	1.023
p = 1.5D	1	1.022	1.025	1.026	1.048
p = 1.8D		1.022	1.025	1.055	1.081

Flow features under the effects of RIBS

With the application of ribs, the flow development process of impingement cooling become more complex. As the vortical structures play a vital role in determining the flow features and the heat transfer performance,^36,37 the flow development and the vortical structures are detailed illustrated and compared in this section to find out the heat transfer enhancement mechanism of applying ribs.

Figure 7 shows the vortical structure distribution in three cases under p = 1.5D (2000 s⁻¹ swirling strength³⁸). For the benchmark case of D_r = 0.0D, two major vortical structures are clearly captured. The first one located at the bottom of the target channel is classified as a-type vortical structure. This kind of vortical structure is induced by the interference between surface flow and cross flow and is more apparent at the upstream region (also see Figure 8). The second one extended from the jet hole outlet is classified as b-type vortical structure. This kind of vortical structure is induced by the interference between cooling jet and cross flow and is less obvious at the upstream region (also see Figure 10). With the application of ribs, both those two kinds of vortical structures become less obvious which indicates that the interferences among cross flow, surface flow and cooling jet is diminished. Besides, a horseshoe shaped vortical structure occurs around the bottom side of ribs and is classified as c-type vortical structure. This kind of vortical structure is induced by the interference between ribs and cross flow and is more apparent at the downstream region (also see Figure 9). With the increase of D_r, the development of the first two types of vortical structures is more disturbed, but the c-type vortical structure is more apparent. As the influence of P on the vortical structure distribution can not be clearly distinguished, the results under p = 1.2D and 1.8D are not shown any more.

Figure 7.

Comparison of vortical structure distribution with a swirling strength of 2000 s⁻¹ (p = 1.5D): (a) benchmark case, (b) D_r = 0.5D, and (c) D_r = 1.0D.

Figure 8.

Static pressure contour and surface streamline on the symmetry plane: (a) benchmark case, (b) p = 1.2D, and (c) p = 1.8D.

Figure 9.

Static pressure contour and surface streamline on the plane of y/D = 0.5.

Figure 8 shows the comparison of P_s distribution as well as the surface streamline on the symmetry plane among different cases. Once the coolant impinges on the target surface, the coolant diffuses in all directions and the surface flow is formed. The surface flow toward the upstream direction collides with the cross flow, and a fountain like flow^39,40 is formed. A pair of a-type vortical structures are generated aside the fountain like flow. As the cross flow accumulates gradually along the channel axial direction, the impinging jet is squeezed and bent, the surface flow that toward the upstream direction is weakened, the fountain like flow and the a-type vortical structure die down, the impingement intensity of cooling jet is then gradually weakened along the channel axis.

With the application of cylindrical ribs, the development processes of both the cross flow and the surface flow are disturbed and obstructed. Under p = 1.2D, the cross flow is obstructed when it is about to impact the downstream jet, the direct flow interference between the surface flow and the upstream cross flow is significantly weakened especially when D_r > 0.25D, the a-type vortical structure is then generated aside the ribs instead of the fountain like flow. With the increase of D_r, the cross flow development is more obstructed, the impinging jet is then less bent by the cross flow which suggests a larger jet impingement intensity, both ΔP_t and $\bar{\bar{Nu}}$ are enhanced consequently. Under p = 1.8D, though the cross flow development is also disturbed, it has enough space to recover back to the center before impacting the downstream jet. Thus, the flow collision as well as the a-type vortical structure can still exist ahead of the impinging jet. Besides, as the ribs are close to their upstream jets, the surface flow that toward the downstream direction is partially resisted and forced back to the upstream direction. In such a way, the flow collision, that is, the obstructing effect of the surface flow on the cross flow, is intensified. The impact strength of the cross flow on the impinging jet is then more countered by the surface flow. With the increase of D_r, the a-type vortical structure is more apparent, especially at the target channel downstream region where the cross flow is more developed. It suggests that more surface flow is forced toward the upstream direction to counter the effect of the cross flow, the impinging jet is then less bent and squeezed. Consequently, with the increase of D_r, the $\bar{\bar{Nu}}$ increases a lot and the PEC also tends to increase.

It can be concluded that the heat transfer enhancement mechanism is influenced by the rib locations. Firstly, the effect of cross flow is more diminished by ribs when the ribs are closer to their downstream jets. Secondly, the surface flow is more forced toward the upstream direction to counter the impact of cross flow when the ribs are closer to their upstream jets. When p = 1.5D, it’s a combination of those two mechanisms, so the relevant results are not discussed any more.

To further explain those two heat transfer enhancement mechanisms, Figure 9 shows the static pressure contour and surface streamline on the horizontal plane of y/D = 0.5. For the benchmark case, there are two high P_s regions near each jet hole. The upstream one is caused by the flow collision, the downstream one represents the core region of the impinging jet. The low P_s region between those two high P_s region is caused by the a-type vortical structure. As shown in the surface streamline, due to the flow collision, part of the upstream cross flow is forced to expand laterally, the downstream cooling jet is then less influenced. With the application of ribs, the cross flow is separated and the c-type vortical structure is generated behind the ribs. When D_r = 0.25D and p = 1.2D, only a small amount of the cross flow can recover back to the center after being separated, the flow collision occurs much closer to the upstream rib, the low P_s region becomes more obvious. When D_r = 0.25D and p = 1.8D, more cross flow can recover back to the center after being separated, the low P_s region is more similar with that of D_r = 0.0D. With the increase of D_r, the flow features are more complex and the pressure loss becomes larger. As the target channel outlet pressure is fixed, the global pressure level inside the target channel is then increased about 50–80 Pa. When D_r = 1.00D and p = 1.2D, the cross flow can hardly recover back to the center after being disturbed, the a-type vortical structure is generally formed when the surface flow impacts the upstream rib. When D_r = 1.00D and p = 1.8D, it is evident that the impinging jet is squeezed by the downstream rib, and the static pressure ahead of each rib becomes much higher, the large reverse pressure gradient pushes more surface flow toward the upstream direction to counter the impact of the cross flow.

Figure 10 shows the static pressure contour and surface streamline on several y–z planes. The development process of the b-type vortical structure along the channel axis can be well reflected by the surface streamline. Refer to the film cooling investigations with inclined coolant injection,⁴¹ the b-type vortical structure may be analogous to the anti-kidney vortex by which the flow losses and the heat convection outside the impingement regions are all increased. As the flow interaction becomes much stronger along the channel axial direction, the b-type vortical structure becomes more pronounced until it occupies the entire channel. With the application of ribs, the development of the cross flow is disturbed, the development of the b-type vortical structure and its effect on the target surface heat transfer are then impaired especially under a larger D_r.

Figure 10.

Static pressure contour and surface streamline on several y–z planes: (a) benchmark case, (b) p = 1.5D, D_r = 0.5D, and (c) p = 1.5D, D_r = 1.0D.

Figure 11 shows the tangential velocity distribution on the symmetry plane. The region where the velocity greater than 10 m·s⁻¹ (except for the impinging jet occupied regions) can be regarded as the region where the b-type vortical structure is well developed. When p = 1.2D, the b-type vortical structure is not obviously influenced by ribs, the region where the velocity greater than 10m·s^-1 occurs at the rear of the fifth jet under all D_r. However, when p = 1.8D, the b-type vortical structure is significantly hindered by ribs, especially under a larger D_r. This is because this anti-kidney shaped vortical structure is induced at the rear of the impinging jet, its development space is taken up when the ribs are located too close to the upstream jet. Besides, when D_r = 0.25D, the b-type vortical structure is not heavily influenced by the rib location. However, when D_r = 1.00D, a smaller P yields to a better development of this anti-kidney shaped vortical structure.

Figure 11.

Tangential velocity distribution on the symmetry plane: (a) p = 1.2D, (b) p = 1.8D, (c) D_r = 0.25D, and (d) D_r = 1.00D.

Impingement heat transfer performance

Figure 12(a)–(c) show the spanwise-averaged Nusselt number $\bar{Nu}$ under different P. For the benchmark case, owe to the initial accumulation of the cross flow, the turbulence level is improved,^9,34 the full development length of impinging jet is shortened, the $\bar{Nu}$ peak value increases gradually till the third impingement region. Owe to the further accumulation of the cross flow, the $\bar{Nu}$ at the impingement region start to decrease from the fourth impingement region. Owe to the excessive accumulation of the cross flow, the impinging jet is significantly bent at the further downstream region, the surface flow that toward the upstream direction can hardly be generated to counter the impact of cross flow, so the $\bar{Nu}$ peak value decreases a lot at the sixth impingement region. After then, due to the enhanced heat convection as a consequence of the more developed cross flow and the b-type vortical structure, the $\bar{Nu}$ peak value tends to increase a little bit till the 10th impingement region. With the application of ribs, the basic variation trend of $\bar{Nu}$ peak value is not significantly changed. However, with the increase of D_r, since the cross flow development is more obstructed, the impingement heat transfer enhancement caused by the initial accumulation of the cross flow is more weakened, the $\bar{Nu}$ at the first five impingement regions decreases gradually. Meanwhile, due to the two heat transfer enhancement mechanisms, the cooling jet is less bent under a larger D_r, the last five $\bar{Nu}$ peak values increase a lot. In such a way, both the impingement heat transfer performance and the heat transfer uniformity along the channel axis are improved. Besides, according to the second heat transfer enhancement mechanism, the surface flow is more forced toward the upstream direction when the ribs are closer to their upstream jets. As there is no rib applied downstream the 10th impinging jet, there is a $\bar{Nu}$ peak value reduction at the 10th impingement region which is more apparent under a larger P.

Figure 12.

Comparison of $\bar{Nu}$ among different D_r and P: (a) p = 1.2D, (b) p = 1.5D, (c) p = 1.8D, and (d) different P.

Figure 12(d) compares the $\bar{Nu}$ curve among different P. When D_r = 0.25D, the differences among three rib locations are not apparent. However, when D_r = 1.00D, the $\bar{Nu}$ peak value increases with the increase of P at the upstream region. This is because the positive effect of the initial accumulation of the cross flow is more diminished under a smaller P. At the downstream region, a larger $\bar{Nu}$ is also obtained under a larger P. This is because the impact strength of a well-developed cross flow on the cooling jet is more reduced by the counter effect of the surface flow than by the direct blocking of ribs. It can be concluded that, the second heat transfer enhancement mechanism is more conducive in improving heat transfer performance on the target surface. In addition, Figure 13 shows the comparison of area-averaged Nusselt number ${\bar{\bar{Nu}}}_{jet}$ for each jet hole cycle (the area between the 1.5D upstream and downstream line of each jet hole) to better show the influence of the ribs. Due to the enhanced heat convection between the cross flow and the target surface along the target channel axis, the ${\bar{\bar{Nu}}}_{jet}$ tends to increase at the target channel downstream region, the other conclusions are similar with that obtained from Figure 12.

Figure 13.

Comparison of ${\bar{\bar{Nu}}}_{jet}$ for each jet cycle.

Figure 14 shows the Nu_D contours under different P. The starting location (red lines) where the high impingement heat transfer area (Nu_D > 110) disappears is marked. In the benchmark case, with the increased influences of the cross flow along the channel axis, the oval shaped high Nu_D region is gradually changed into a crescent shape, the impingement heat transfer deteriorates abruptly at the downstream of the fifth impingement region due to the overbent impinging jet. The heat convection outside the impingement region is increased due to the more accumulated cross flow and better developed b-type vortical structure. With the application of ribs, the high heat transfer area can last to a more downstream region as a consequence of the two heat transfer enhancement mechanisms. When p = 1.2D, due to the direct resistance of ribs, the impingement heat transfer is improved even under a smaller D_r. The shape of the impingement region at the target channel downstream area is more like a V-shaped boomerang with the increase of D_r. When p = 1.8D, more surface flow is forced toward the upstream direction, the flow collision intensity is enhanced, the impingement heat transfer improvement is more apparent under a larger D_r. The impingement region at the target channel downstream area is more like a fuller meniscus with the increase of D_r. The relevant result when p = 1.5D is somewhere in between. Besides, under all three P, the impingement region can be extended to a further downstream area than the benchmark case, the low heat transfer area between adjacent impingement region is then compressed. This phenomenon is more apparent under a larger D_r. Therefore, the $\bar{Nu}$ valley values shown in Figure 12 are much larger under a larger D_r, especially at the downstream region.

Figure 14.

Comparison of Nu_D distribution on the target surface: (a) p = 1.2D, (b) p = 1.5D, and (c) p = 1.8D.

Conclusions

The impingement heat transfer enhancement mechanism of cylindrical ribs applied between adjacent jets inside a semicircular target channel has been numerically studied. The influences of rib diameter and rib to jet hole distance are also investigated. Major vortical structures coupled with pressure and velocity distributions are illustrated and analyzed to intuitively show the effects of ribs. The heat transfer characteristics is then discussed and compared accordingly. The key conclusions of this paper are listed as follows:

(1) With the application of ribs, the impinging jet is less affected by the cross flow at the cost of a more complex flow features inside the target channel, the total pressure drop increases about 5.7%–9.1%, the area-averaged Nusselt number increases about 4.3%–11.3% and the maximum Performance Evaluation Criterion (PEC) enhancement is about 2.1%.

(2) When pursuing a smaller flow loss, a smaller rib diameter coupled with a smaller rib to downstream jet distance are preferred. when pursuing a larger heat transfer enhancement and larger PEC, a larger rib diameter coupled with a larger rib to downstream jet distance are preferred.

(3) The heat transfer enhancement mechanism of applying ribs is affected by the rib to jet distance. Firstly, the effect of cross flow is more directly diminished by ribs when the ribs are closer to their downstream jets. Secondly, the surface flow is more forced toward the upstream direction to counter the impact of cross flow when the ribs are closer to their upstream jets.

(4) With the increase of rib diameter, the heat transfer performance at the target channel upstream region decreases gradually and the heat transfer performance at the target channel downstream region increases gradually. Thus, both the heat transfer performance and the uniformity of the heat transfer distribution are improved.

(5) The impingement region can extend to a further downstream area with the application of ribs, the low heat transfer area between adjacent impingement regions is then compressed, especially under a larger D_r. When the ribs are much closer to their upstream jets, the impingement heat transfer enhancement is more pronounced.

In this paper, the influencing mechanism of the location and diameter of the circular rib on impingement heat transfer performance is investigated, however, many factors, such as the effect of rib shape, target surface effusion hole, Reynolds number and so on, are not taken into considerations. Besides, a simplified semicircular leading edge model is used, the effect of actual vane structure and the heat conduction are also not studied. Thus, a more comprehensive study is still needed to investigate the influence of ribs to further verify its heat transfer enhancement mechanism as well as its practicability at the leading edge impingement cooling.

Footnotes

Appendix

Handling Editor: Ahmed Al-Sammarraie

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research work was funded by the Young Elite Scientists Sponsorship Program by CAST (Program No. 2022QNRC001) and the Youth Fund project of Natural Science Foundation of Jiangsu Province (Funding No. BK20210301), Open Fund of Key Laboratory of Inlet and Exhaust System Technology of Ministry of Education (Funding No. CEPE2024013).

ORCID iD

Junfei Zhou

References

Dutta

Singh

Opportunities in jet-impingement cooling for gas-turbine engines. Energies 2021; 14: 6587.

Plant

Friedman

Saghir

MZ.

A review of jet impingement cooling. Int J Thermofluids 2023; 17: 100312.

Han

Goldstein

RJ.

Jet-impingement heat transfer in gas turbine systems. Ann N Y Acad Sci 2001; 934: 147–161.

Forster

Weigand

Experimental and numerical investigation of jet impingement cooling onto a concave leading edge of a generic gas turbine blade. Int J Therm Sci 2021; 164: 106862.

Tepe

AÜ

. Numerical investigation of a novel jet hole design for staggered array jet impingement cooling on a semicircular concave surface. Int J Therm Sci 2021; 162: 106792.

Wright

Ahmed

Yan

, et al. Experimental and numerical heat transfer investigation of impingement jet nozzle position in concave double-wall cooling structures. Heat Transf Eng 2022; 43: 1108–1118.

Pachpute

Premachandran

Turbulent multi-jet impingement cooling of a heated circular cylinder. Int J Therm Sci 2020; 148: 106167.

Tepe

AÜ

Yetişken

Uysal , et al. Experimental and numerical investigation of jet impingement cooling using extended jet holes. Int J Heat Mass Transf 2020; 158: 119945.

Lee

Ren

Ligrani

, et al. Crossflows from jet array impingement cooling: hole spacing, target plate distance, Reynolds number effects. Int J Therm Sci 2015; 88: 7–18.

10.

Yang

Qiu

, et al. Temperature uniformity characteristics of array jet impingement cooling with the maximum cross-flow scheme. Int J Therm Sci 2023; 187: 108161.

11.

Otero-Pérez

Sandberg

Mizukami

, et al. High-fidelity simulations of multi-jet impingement cooling flows. J Turbomach 2021; 143: 081011.

12.

Chen

Liu

Rao

, et al. Numerical investigation on conjugate heat transfer of impingement/effusion double-wall cooling with different crossflow schemes. Appl Therm Eng 2019; 155: 515–524.

13.

Uysal

Chyu

, et al. Heat transfer on internal surfaces of a duct subjected to impingement of a jet array with varying jet hole-size and spacing. J Turbomach 2006; 128: 158–165.

14.

Terzis

Ott

Cochet

, et al. Effect of varying jet diameter on the heat transfer distributions of narrow impingement channels. J Turbomach 2015; 137: 021004.

15.

Singh

Ekkad

, et al. Effect of crossflow regulation by varying jet diameters in streamwise direction on jet impingement heat transfer under maximum crossflow condition. Numer Heat Transf A Appl 2017; 72: 579–599.

16.

Zhou

Tian

, et al. Numerical investigation on flow and heat transfer characteristics of single row jet impingement cooling with varying jet diameter. Int J Therm Sci 2022; 179: 107710.

17.

Ligrani

Click

Ritchie

, et al. Double wall cooling of an effusion plate with cross flow and impingement jet combination internal cooling: comparisons of main flow contraction ratio effects. Int J Heat Mass Transf 2020; 150: 119196.

18.

Chi

Liu

Zang

Geometrical optimization of nonuniform impingement cooling structure with variable-diameter jet holes. Int J Heat Mass Transf 2017; 108: 549–560.

19.

Deng

Xiao

, et al. Heat transfer enhancement of impingement cooling by different crossflow diverters. J Heat Transf 2022; 144: 042001.

20.

Salem

Soliman

Amano

RS.

Heat Transfer and crossflow investigations for jet impingement cooling applications. Int J Gas Turbine Propul Power Syst 2024; 15: 15–23.

21.

Salem

Soliman

Amano

. Crossflow minimization for jet impingement cooling applications, ASME paper, GT2023-101162, 2023.

22.

Deng

Feng

Heat transfer enhancement of impingement cooling with corrugated target surface. Int J Therm Sci 2022; 171: 107251.

23.

Madhavan

Ramakrishnan

Singh

, et al. Jet impingement heat transfer enhancement by U-shaped crossflow diverters. J Therm Sci Eng Appl 2020; 12: 041005.

24.

Ren

Yang

, et al. Experimental investigation of micro cooling units on impingement jet array flow pressure loss and heat transfer characteristics. Energies 2021; 14: 4757.

25.

Yalçınkaya

Durmaz

Numerical analysis of jet impingement cooling with elongated nozzle holes on a curved surface roughened with V-shaped ribs. J Therm Anal Calorim 2024; 149: 3453–3470.

26.

Kim

Jung

Bang

, et al. Heat transfer measurements for array jet impingement with castellated wall. J Turbomach 2022; 144: 031009.

27.

Sun

Luan

, et al. The heat transfer enhancement mechanism of W-shaped micro-ribs on impingement cooling. Appl Therm Eng 2024; 252: 123658.

28.

Fathi

Nejat

Conjugate heat transfer investigation of impingement cooling for ribbed internal passage of a turbine vane. Int J Therm Sci 2022; 178: 107589.

29.

Shukla

Dewan

Singh

, et al. Heat transfer characteristics of jet impingement on a surface mounted with ribs using LES. Int J Therm Sci 2023; 187: 108210.

30.

Yang

Ren

Jiang

. Experimental and numerical investigation of impingement cooling with spent flow in the blade leading edge. In: Proceedings of 4th international symposium on jet propulsion and power engineering, ISJPPE-2012-A0127.

31.

Yang

Ren

Jiang

, et al. Experimental and numerical investigation of unsteady impingement cooling within a blade leading edge passage. Int J Heat Mass Transf 2014; 71: 57–68.

32.

Roache

PJ.

Verification and validation in computational science and engineering. Vol. 895. Albuquerque, NM: Hermosa, 1998.

33.

Florschuetz

Metzger

Truman

CR.

Jet array impingement with crossflow - correlation of streamwise resolved flow and heat transfer distributions, NASA Contractor Report 3373, Department of Mechanical Engineering, Arizona State University, Tempe, January 1981.

34.

Llucià

Terzis

Ott

, et al. Heat transfer characteristics of high crossflow impingement channels: effect of number of holes. Proc IMechE 2015; 229: 560–568.

35.

Fechter

Terzis

Ott

, et al. Experimental and numerical investigation of narrow impingement cooling channels. Int J Heat Mass Transf 2013; 67: 1208–1219.

36.

Ligrani

Oliveira

Blaskovich

Comparison of heat transfer augmentation techniques. AIAA J 2003; 41: 337–362.

37.

Ligrani

Heat transfer augmentation technologies for internal cooling of turbine components of gas turbine engines. Int J Rotating Machinery 2013; 2013: 1–32.

38.

ANSYS Fluent. ANSYS fluent theory guide 15.0. Canonsburg, PA: ANSYS Inc, 2013.

39.

San

Lai

MD.

Optimum jet-to-jet spacing of heat transfer for staggered arrays of impinging air jets. Int J Heat Mass Transf 2001; 44: 3997–4007.

40.

Terzis

On the correspondence between flow structures and convective heat transfer augmentation for multiple jet impingement. Exp Fluids 2016; 57: 146.

41.

Bunker

RS.

A review of shaped hole turbine film-cooling technology. J Heat Transf 2005; 127: 441–453.