Introduction

Studying jet flows is greatly important in both scientific research and industrial applications. These types of flow are frequently observed in different industrial applications, including cooling and mixing processes, fuel injection, and aerospace propulsion systems. Jet flows are investigated in scientific research to study turbulence, which aids in understanding complex flows. This information is important for improving energy efficiency, reducing pollutant emissions as well as boosting jet engine effectiveness. Jet flow is necessary in industrial applications for tasks such as combustion, regulating flow, and jet cutting processes. Turbulence phenomenon in jet flows intensifies heat transfer and mixing phenomena, essential for cooling systems and heat exchangers. In summary, studies on jets result in significant advancements in both theory and practical applications.

Among the problems that have attracted the attention of scientists is the problem of impinging jet arrays, particularly the interaction of such jets, which is the focus of the current study due to their wide variety of applications in the industrial environment, such as homogenizing atmospheres to achieve thermal comfort in air conditioning, equalizing temperatures in furnaces, ensuring good mixing in burners, and improving combustion in engines with a view to achieving clean propulsion and reducing unnecessary energy waste. Many researchers conduct parametric experiments and numerical studies on impinging jet arrays. These investigations aim to identify the various factors that can influence the dynamic and thermal characteristics of this kind of complex turbulent flow.

As mentioned in the recent review work of Barbosa et al.1, jet arrangement is one of the most important factors that influence the heat transfer mechanism as well as the fluid flow2. Another important factor is the jet inclination, which refers to the angle at which the jets impact the impinging plate3. Similarly, the shape of the nozzle outlet, the distance between individual jets within the array, and the distance between the jet nozzle and the impinging plate all significantly alter the flow behavior4,5,6. The initial temperature of the impinging plate and the jet velocity itself have a substantial impact on heat transfer, in addition to the jet configuration7,8. Dimensionless characteristics such as the Prandtl number and Mach number are also significant determinants9,10. Another key factor influencing the process is the well-known Reynolds number11,12, which represents flow characteristics and turbulence potential. The impingement process may also be influenced by the impinging plate itself, as its curvature, surface roughness, and even its inclination angle with respect to the nozzle plate can all be important13. Lastly, further complexity in the impinging jet array is added by the degree of turbulence at the jet exit and the behavior of the flow field when the impinging plate is moving14.

Youn et al.15 studied the heat transfer characteristics of impinging jets array with effusion holes, providing important information for the design of cooling systems for gas turbine casings. The effects of hole pitch, separation between jet and impinging plates, Reynolds number based on the hole diameter, and hole diameter itself were investigated using a three-dimensional numerical unit cell model that included k–ω SST turbulence model and transition model. These parameters are respectively ranged between 1–10, 6–20, and 3500–35,000. The study proposed a new correlation to estimate the mean heat transfer along the impinging plate after analyzing the results of local and mean Nusselt numbers through 74 simulation cases. Results show how these parameters have a major influence on the cooling system’s heat transfer properties. Brakmann et al.16 experimentally and numerically investigate how impinging jets and detached rib turbulators can work together to cool turbomachinery industrial applications. This work focuses on the target wall with detached ribs and a 9 × 9 jet whole array on a target plate. The investigation looks at the effects of dimensionless rib spacing (in range 0.3–0.08), jet Reynolds numbers (in the ranges 15,000–35,000), and dimensionless separation distances (in the ranges 3–5). The TLC (Transient Liquid Crystal) method is used to evaluate heat transfer, and a CFD model that uses the SST turbulence model is used to perform detailed numerical flow analysis. Detached ribs clearly change the field of flow when compared with smooth targets. It is found that ribs can increase the global Nusselt number by up to 4% and decrease the relative discharge coefficient by up to 11%.

Hayee et al.17 explores the heat transfer behavior of an impinging jet array under fully developed flow conditions. The study uses a 5 × 5 array of jets and each jet is discharged from pipe nozzles with a diameter of 17.2 mm and a length of 300 mm arranged in an in-line pattern. The investigation mentioned varying the non-dimensional jet–jet spacing (S) and the jet-plate distance (H) with S = 4, 6, and 8, and H = 2, 4, 6, and 8. The Reynolds number at the jet’s outlet is set at 10,000, 20,000, 30,000, and 40,000. The temperature distributions along the impinging plate are measured using an infrared camera, and the Nusselt numbers are calculated based on these measurements. The findings reveal that the maximum average Nusselt number is achieved at H = 4 for all jet–jet spacing. A heat transfer correlation is developed based on this peak average Nusselt number at H = 4, contrasting with previous findings for impinging jet arrays from orifice nozzles, where average heat transfer decreases with increasing jet-plate distance. Xing et al.18 performed an experimental and numerical study on the thermal characteristics of impinging jet array. The experimental setup guarantees the measure of local jet temperatures on an impinging plate using a (TLC) transient liquid crystal technique. The investigation concentrated on how the local Nusselt numbers and pressure loss can be affected by various impingement plate patterns, jet-plate distance, crossflow schemes, and jet Reynolds numbers. The main objective of the numerical analysis was to confirm that the calculated heat transfer rates were accurate. The findings demonstrated a strong correlation between the computational and experimental data, with precise predictions of the local heat transfer coefficients. The study concludes that those impinging jet array setups thermal design process can widely benefit from the application of CFD simulations.

Culun et al.19 investigates the combined heat transfer effects of impinging jets array. The jet shape and arrangement, jet density in both spanwise and streamwise directions, and confinement type are the design parameters that affect the flow. ANSYS-CFX, a commercial CFD program, was used to conduct numerical analysis. Heat transfer is measured using the average Nusselt number. According to the results obtained, heat transfer in the densest jet array inside a single exit confined channel behaves similarly to a single jet because of the strong interference from neighboring jets. On the other hand, jet arrangement is found to have the least effect on heat transfer and square jets produced the highest heat transfer. Aldabbagh and Sezai20 performed numerical analysis along the laminar region for finite values of Reynolds numbers and jet–jet spacing. The authors discussed the three-dimensional numerical simulation of heat transfer characteristics of a 3 × 3 arrangement of square jets in spent fluid clearing. The purpose of this study is to find out how a very small jet-plate distance (H = 0.25) affects the removal of spent air and local Nusselt number. The explanation is that, while quite a distance H = 0.25 is optimal for the attachment of a wall jet to the impinging plate caused by fluid redirected to the spent air hole, redirecting the fluid causes wall jets to detach due to the spent air hole. However, at a larger nozzle to plate distance (H = 2), the removal of spent fluid in the plate increased the heat transfer rate when jet–jet spacing S < 4. The impact of spent fluid removal is more pronounced for smaller jet–jet spacing because of increased jet interactions and crossflow effects.

San and Lai21 investigated the heat transfer characteristics of impinging jet arrays in different configurations concentrating on factors such as jet fountains and interference. By using experimental data, optimal jet–jet spacing to diameter ratios for various jet array configurations were established with curve fitting equations provided to correlate these findings. The authors offer insight into heat transfer mechanisms involved in this kind of flow and some guidelines for better design of the heat exchanger systems. The authors found that for a small Reynolds number (Re = 10,000), if the jet-plate distance (H) is large (≥ 3.0), the strength of the fountain is weak, causing the second relative maximum of the Nusselt number at the stagnation point of the central jet to disappear. For small H values (< 3.5), the shear layer expansion in the jet is minor, and jet interference before impingement should occur at a small jet–jet spacing. In this case, due to the limitations of the experimental apparatus, the first relative maximum of the stagnation point Nusselt number cannot be observed. From the results of this experiment, for Re in the range of 10,000–30,000 and S in the range of 4–16, the optimal S for H = 2, 3, and 5 is 8, 12, and 6, respectively. San and Chen22 conduct an experimental setup to examine thermal behavior of turbulent impinging jet arrays by varying the jet-plate distance (H) in the range 0.5–3, as well as the jet–jet spacing (S) in the range 2–8. The Reynolds number for the jets was 20,000. At low S and H values (S = 2 and H = 0.5), a peak Nusselt number (Nu) was observed due to strong jet interactions on the impinging plate, particularly between the central jet and its neighbors. As S and H increased, the interaction between jets diminished. For small S and large H values (S = 2 and H ≥ 2), pre-impingement jet interference reduced heat transfer but resulted in a uniform Nusselt number distribution within the jet array’s coverage area. At intermediate S and large H values (S = 4 and H ≥ 2), both jet interaction and interference were minimal, but the cross flow from the central jet reduced the heat transfer efficiency of the neighboring jets. When S values were large (S ≥ 6), regardless of H (within 0.5–3), each jet maintained an independent cooling zone on the impingement plate. The findings also indicated that the maximum Nusselt number variation under the jet array increased almost linearly with S.

The experimental studies of heat transfer from impinging gas jet array to flat surfaces are the main topic of Katti and Prabhu23. Various jet array configurations were investigated, the configuration with a spanwise pitch of four times the nozzle diameter showed the best results in terms of pressure loss coefficient and average Nusselt number. The impact of spanwise pitch on the stagnation point heat transfer coefficient, crossflow characteristics, and local heat transfer distribution was examined in this study. The findings indicated that because of crossflow and jet-to-jet interaction, configurations with a smaller spanwise pitch had lower stagnation point heat transfer coefficients. Heat transfer coefficients and strip-wise average Nusselt numbers were predicted using correlations that were generated as functions of the crossflow velocity, jet velocity, and Reynolds number. The research sheds light on the dynamics of heat transfer in impinging jet array. Lee et al.24 conducted a study that examined how array spacing, Reynolds number, and jet-plate distance impact both crossflows and heat transfer in an impinging jet array. These authors discussed the interrelationships between impinging jets and crossflows leading to changes in heat transfer especially at further downstream distances. This investigation consisted of experiments setup which allow calculating the local Nusselt numbers via thermocouples and infrared imaging. It is found that locally increased Nusselt numbers are the outcome of the impingement crossflow. Due to stronger jet interactions, increased local mixing and turbulent transport magnitudes, and greater downstream locations, these changes most frequently occur. This happens in channels at lower Reynolds numbers because impingement jets are constrained by smaller jet-plate distances and array spacing. Li et al.25 examines the dynamic characteristics of impinging jet array with significant impact angles on a flat plate. The core objective of this study is to investigate the mutual interaction between these jets and heat transfer with the help of three-dimensional mathematical modeling as well as experimental oil flow visualization. Among the variables considered in these studies include Reynolds number, jet–jet spacing, and distance from the plate which influence flow patterns and heat transfer. On the one hand, grooves were formed on the impinging plate because of mutual entrainment due to Coanda effect, resulting in irregular grooves from conflicting impingements downstream. Near the stagnation point, the wall shear stress peaks in the near field, however, in downstream regions it drops drastically. Heat transfer behavior is significantly influenced by Reynolds number, jet–jet spacing, and jet-plate distance while flow structure shows only little sensitivity to Reynolds number because it depends on how flows interact especially their impact at impingement surface.

The previous mentioned works15,16,17,18,19,20,21,22,23,24,25 as well as the quick summary of the several factors that can affect the turbulent flow of impinging jet array1,2,3,4,5,6,7,8,9,10,11,12,13,14 clearly shows the absence of any investigation on the number of jets effect, this parameter means the number of nozzles that allow the flow to be discharged. The increase in the number of jets raises an obvious variable: the spacing between these jets. This is remarkably interesting, motivating us to numerically study the jet–jet spacing (S) effect and combine it with the number of jets (N). Furthermore, the correlations found in the literature that predict the average Nusselt number are generally given as a function of one or two parameters. Thus, the goal of our study is to provide empirical correlation for the average Nusselt number based on three parameters: the number of jets (N), the distance between the jets (S), and the distance between the jets and the plate (H).

The top priority of our study lies in innovative and comprehensive strategies. It considers several parameters simultaneously to evaluate the heat transfer phenomenon. This approach provides a new perspective and useful tools for researchers and industry experts. This allows for more accurate and appropriate predictions of jet array characteristics. In this context, our computational fluid dynamics (CFD) study focuses on investigating the mutual effect of the jet–jet spacing and the number of jets on the thermal characteristics and flow dynamic of impinging jet array. We also focus specifically on investigating how the effect of jet–jet spacing changes with the number of jets and vice versa. By providing a deeper understanding of the fundamental relationship between these factors and their impact on this kind of flow, our article provides important support for scientists’ research efforts.

Geometric configuration

The geometric configuration of the present work (Fig. 1) is used to analyze the flow field generated by impinging jet array on a heated impinging plate. Different jet numbers (N) are considered, such as N = 5, 7, 9, and 11 (Fig. 1 is just a representative case of N = 5). The nozzles are identical, rectangular and each has a thickness w = 3.175mm and a width l = 152.4mm and is aligned laterally (along the x-axis). The distance between the ejection nozzles and the impinging plate (called the jet-plate distance) is denoted (h) and is constant and equal to 6 times the nozzle thickness (h = 6 × w), giving a dimensionless jet-plate distance H = h/w = 6. Note that this fixed value of the jet-plate distance value is the same used by the experimental study of Gardon and Akfirat26 on which we have validated our numerical model.

Fig. 1
figure 1

Geometric configuration of impinging jets array.

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During this numerical simulation, four jet–jet spacing (s) are considered, namely s = 1 × w, 2 × w, 3 × w and 4 × w, giving dimensionless spacing S = s/w = 1, 2, 3 and 4. The impinging plate is subjected to a constant temperature and its length is fixed (L = 80 × w). The dimensions of the numerical model are then 6 × w and 80 × w, as clearly shown in Fig. 2 along the axial (y-axis) and lateral (x-axis) directions, respectively. As previously noted, our numerical model is first validated on the geometric configuration of Gardon and Akfirat26 in the case of a single impinging jet (with a number of jets N = 1). In this configuration, the length of the impinging plate for the validation step is L = 48 × w. The extension of the impinging plate length in the rest of the simulations to L = 80 × w is conducted to ensure good visualization of the impinging jet array flow, which will be more expanded laterally (along the x-axis) when the number of jets (N) is increased.

Fig. 2
figure 2

Domain dimensions, mesh construction and boundary conditions.

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Conservation equations and turbulence modeling

The governing equations system adopted in the present numerical investigation is assumed to be steady and incompressible. Furthermore, the nozzle length (l = 152.4 mm) is significantly larger than its width (3.175 mm), resulting in an aspect ratio of 48. This allows us to approximate the flow as bidimensional (2D). The Reynolds-averaged Navier–Stokes Equations (RANS), including continuity, momentum, and energy equations, are written, in Cartesian coordinates as follows:

$$\frac{\partial }{{\partial {\text{x}}_{{\text{i}}} }}\left( {\uprho {\text{u}}_{{\text{i}}} } \right) = 0$$
(1)
$$\frac{\partial }{{\partial {\text{x}}_{{\text{j}}} }}\left( {\uprho {\text{u}}_{{\text{i}}} {\text{u}}_{{\text{j}}} } \right) = - \frac{{\partial {\text{p}}}}{{\partial {\text{x}}_{{\text{i}}} }} + \frac{\partial }{{\partial {\text{x}}_{{\text{j}}} }}\left[ {\upmu \left( {\frac{{\partial {\text{u}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{j}}} }} + \frac{{\partial {\text{u}}_{{\text{j}}} }}{{\partial {\text{x}}_{{\text{i}}} }} - \frac{2}{3}\updelta _{{{\text{ij}}}} \frac{{\partial {\text{u}}_{{\text{l}}} }}{{\partial {\text{x}}_{{\text{l}}} }}} \right)} \right] + \frac{\partial }{{\partial {\text{x}}_{{\text{j}}} }}\left( { -\uprho \overline{{{\text{u}}_{{\text{i}}}^{\prime } {\text{u}}_{{\text{j}}}^{\prime } }} } \right)$$
(2)

The energy equation is given by the following:

$$\frac{\partial }{{\partial {\text{x}}_{{\text{i}}} }}\left[ {{\text{u}}_{{\text{i}}} \left( {\uprho {\text{E}} + {\text{p}}} \right)} \right] = \frac{\partial }{{\partial {\text{x}}_{{\text{j}}} }}\left( {\uplambda _{{{\text{eff}}}} \frac{\partial T}{{\partial {\text{x}}_{{\text{j}}} }} + {\text{u}}_{{\text{i}}} \left( {\tau_{{{\text{ij}}}} } \right)_{{{\text{eff}}}} } \right) + {\text{S}}_{{\text{h}}}$$
(3)

The E term is the total energy, λeff is the effective thermal conductivity and (τij)eff is the deviatoric stress tensor defined by:

$$\left( {\uptau _{{{\text{ij}}}} } \right)_{{{\text{eff}}}} =\upmu _{{{\text{eff}}}} \left( {\frac{{\partial {\text{u}}_{{\text{j}}} }}{{\partial {\text{x}}_{{\text{i}}} }} + \frac{{\partial {\text{u}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{j}}} }}} \right) - \frac{2}{3}\upmu _{{{\text{eff}}}} \frac{{\partial {\text{u}}_{{\text{k}}} }}{{\partial {\text{x}}_{{\text{k}}} }}$$
(4)
$$\uplambda _{{{\text{eff}}}} =\uplambda + \frac{{{\text{C}}_{{\text{p}}}\upmu _{{\text{t}}} }}{{\Pr_{{\text{t}}} }}$$
(5)

where λ, Cp and Prt (constant value 0.85) are the thermal conductivity, heat capacity and turbulent Prandtl number, respectively. The Boussinesq hypothesis which essentially relates to the Reynolds stress terms present in Eq. (2) with the mean velocity gradients is adopted for the turbulence closure as presented in Eq. (6):

$$- \overline{{\uprho {\text{u}}_{{\text{i}}}^{\prime } {\text{u}}_{{\text{j}}}^{\prime } }} =\upmu _{{\text{t}}} \left( {\frac{{\partial {\text{u}}_{{\text{i}}} }}{{\partial {\text{x}}_{{\text{j}}} }} + \frac{{\partial {\text{u}}_{{\text{j}}} }}{{\partial {\text{x}}_{{\text{i}}} }}} \right) - \frac{2}{3}\left( {\uprho {\text{k}} +\upmu _{{\text{t}}} \frac{{\partial {\text{u}}_{{\text{k}}} }}{{\partial {\text{x}}_{{\text{k}}} }}} \right)\updelta _{{{\text{ij}}}}$$
(6)

The turbulent kinetic energy (k) and the rate of specific dissipation of turbulent kinetic energy (ω) are calculated from Eqs. (7–8), respectively:

$$\frac{\partial }{{\partial {\text{x}}_{{\text{i}}} }}\left( {\uprho {\text{ku}}_{{\text{i}}} } \right) = \frac{\partial }{{\partial {\text{x}}_{{\text{j}}} }}\left( {\Gamma_{{\text{k}}} \frac{{\partial {\text{k}}}}{{\partial {\text{x}}_{{\text{j}}} }}} \right) + {\text{G}}_{{\text{k}}} - {\text{Y}}_{{\text{k}}} + {\text{S}}_{{\text{k}}}$$
(7)
$$\frac{\partial }{{\partial {\text{x}}_{{\text{i}}} }}\left( {{\uprho \omega }{\text{u}}_{{\text{i}}} } \right) = \frac{\partial }{{\partial {\text{x}}_{{\text{j}}} }}\left( {\Gamma_{\upomega } \frac{{\partial\upomega }}{{\partial {\text{x}}_{{\text{j}}} }}} \right) + {\text{G}}_{\upomega } - {\text{Y}}_{\upomega } + {\text{S}}_{\upomega }$$
(8)

In Eqs. (7) and (8), the terms Gk and Gω denote the generation of k and ω, respectively. Γk and Γw correspond the effective diffusivity for k and ω, respectively. Yk and Yω represent the dissipation of k and ω, respectively; Sk and Sω being the source terms. Γk and Γw are given by the following expressions:

$$\Gamma_{{\text{k}}} =\upmu + \frac{{\upmu _{{\text{t}}} }}{{\upsigma _{{\text{k}}} }}$$
(9)
$$\Gamma_{\upomega } =\upmu + \frac{{\upmu _{{\text{t}}} }}{{\upsigma _{\upomega } }}$$
(10)
$$\upmu _{{\text{t}}} = \upalpha ^{*} \frac{{\uprho {\text{k}}}}{\upomega }$$
(11)

Here, σk = σω denotes the turbulent Prandtl numbers, respectively for k and ω. α* is calculated as follows:

$$\upalpha ^{*} = \upalpha _{\infty }^{*} \left( {\frac{{\upalpha ^{*}_{0} + {\text{Re}}_{{\text{t}}} /{\text{R}}_{{\text{k}}} }}{{1 + {\text{Re}}_{{\text{t}}} /{\text{R}}_{{\text{k}}} }}} \right)$$
(12)
$${\text{Re}}_{{\text{t}}} = \frac{{\uprho {\text{k}}}}{{\upmu \upomega }}$$
(13)
$$\upalpha ^{*}_{0} = \frac{{\upbeta _{{\text{i}}} }}{3}$$
(14)

Here, Rk = 6 and βi = 0.072 are constant, and α* = α*∞ = 1 for k-ω model with high Reynolds number. The terms Gk and Gω respectively in Eqs. (7) and (8) can be obtained by:

$${\text{G}}_{{\text{k}}} = -\uprho \overline{{{\text{u}}_{{\text{i}}}^{\prime } {\text{u}}_{{\text{j}}}^{\prime } }} \frac{{\partial {\text{u}}_{{\text{j}}} }}{{\partial {\text{x}}_{{\text{i}}} }}$$
(15)
$${\text{G}}_{{\text{k}}} =\upmu _{{\text{t}}} {\text{S}}^{2}$$
(16)
$${\text{S}} \equiv \sqrt {2{\text{S}}_{{{\text{ij}}}} {\text{S}}_{{{\text{ij}}}} }$$
(17)
$${\text{G}}_{\upomega } = \upalpha \frac{\upomega }{{\text{k}}}{\text{G}}_{{\text{k}}}$$
(18)
$$\upalpha = \frac{{\upalpha _{\infty } }}{{\upalpha ^{*} }}\left( {\frac{{\upalpha _{0} + {\text{Re}}_{{\text{t}}} /{\text{R}}_{\upomega } }}{{1 + {\text{Re}}_{{\text{t}}} /{\text{R}}_{\upomega } }}} \right)$$
(19)

Here, Rω = 2.95, α∞ = 0.52 and α0 = 1/9 are constant. α* and Ret are respectively given by (Eqs. (12) and (13)) with α = α* = 1 for k-ω model with high Reynolds number27.

Mesh construction, boundary and initial conditions

Mesh construction and boundary conditions

As clearly shown in Fig. 2, the mesh adopted in the present configuration is uniform with a spacing a = 0.08. This gives a total number of meshes equal to 75,000 quadratic cells along the whole domain of dimension 6 × w and 80 × w. Concerning the boundary conditions, at the nozzle’s outlet, the velocity adopts a uniform profile. This is reflected by the “VELOCITY INLET” boundary condition. This boundary condition is derived from the flow characteristics by theories or experimental data. It assumes that the inlet’s velocity distribution is known and used to describe an inflow boundary where fluid enters the computational domain. At the nozzle ejection (y = 0), and along the entire length of the impinging plate (y = 6 × w), the velocity is zero, which translates into the “WALL” boundary condition. The velocity at the wall is zero because there should not be any flow across this barrier. The no-slip condition is also taken into consideration for “WALL” boundary condition, provided that the fluid velocity at the wall is equal to the wall’s velocity which is set to zero in the present configuration. “WALL” condition assumes that viscous effects predominate close to the impinging plate and the nozzle plate. The “PRESSURE OUTLET” boundary condition is applied when the pressure gradient at the flow outlet and on both sides of the impact wall (x =  + − 40 × w) is considered constant. The “PRESSURE OUTLET” condition applies a pressure profile based on experimental data or assumptions, and usually sets the static pressure to a known value, assuming that the flow exits the domain. To obtain the desired pressure distribution at the flow exit, the flow is permitted to modify its velocity and other characteristics due to this boundary condition.

Initial Conditions

At the nozzle ejection (y = 0), the Reynolds number is constant (Re = 5500), resulting in an outlet velocity equal to 25.1 m/s. The calculation of this velocity is based on the hydraulic diameter Dh = 6.2mm. The turbulence intensity considered is I = 2.5%. The impinging plate is heated to a constant temperature of Tw = 320K. The jet temperature at the nozzles ejection and the ambient temperature are such that T0 = Tamb = 300K. All these conditions are the same as being adopted in experimental work of Gardon and Akfirat26, on which our numerical model has been validated. At the nozzle outlet, the turbulent kinetic energy (k0) and its specific dissipation rate (ω0) are given by the following equations:

$${\text{k}}_{0} = \frac{3}{2}\left( {{\text{Iu}}_{0} } \right)^{2} = 0.591\,{\text{m}}^{2} /{\text{s}}^{2}$$
(20)
$$\upomega _{0} = \frac{{{\text{k}}_{0}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{{\text{C}}_{\upmu }^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} 0.07{\text{D}}_{{\text{h}}} }} = 3233\,1/{\text{s}}$$
(21)

The transport equations related to these boundaries and emission conditions were resolved by utilizing the Computational Fluid Dynamics solver, Ansys Fluent, and the finite volume approach. Within the calculated geometrical domain, a finite number of sub-regions (Control Volume) were created. Following their discretization using the second order “UPWIND” scheme, the equations were integrated in each of these control volumes. The “SIMPLEC” algorithm was also selected for the velocity–pressure coupling. Furthermore, when the normalized residuals lie below 10−5, convergence is guaranteed. Additionally, we have confirmed that the simulation results are unaffected by lowering this convergence requirement.

Validation of the numerical model

Mesh sensitivity test

Figure 3a shows the prediction of the local Nusselt number (Nus) at the stagnation point (x = 0, y = 0) as a function of the jet-plate distance (H) for the same conditions adopted in the experimental work of Gardon and Akfirat26. These results are shown for three mesh sizes: Mesh (1) = 75,000, Mesh (2) = 100,000 and Mesh (3) = 150,000 quadratic cells. It is clear from Fig. 3a that the local Nusselt predictions with mesh sizes (2) and (3) are almost identical and present a strong consistency with the experimental results26. However, the prediction of local Nusselt by mesh (1) shows some discrepancy with the same experimental results considered. This clearly shows that mesh (2) is the most suitable for obtaining results in satisfactory agreement with those obtained by Gardon and Akfirat26. So, the mesh size (2) is smaller than the mesh size (3), which reduces the computation time required for simulation. To better validate our simulations and guarantee that our numerical model can predict the flow’s dynamic characteristics, we present a second validation of our numerical results in Fig. 3b using the experimental profile of the dimensionless static pressure (P) from the same experimental study of Gardon and Akfirat26. For this validation, the considered Reynolds number is Re = 5500, the number of jets N = 3, the jet–jet spacing S = 15, and the jet-plate distance H = 4. The previously chosen mesh 2, which has 100,000 quadratic cells, is utilized. A good agreement between our numerical results and those of Gardon and Akfirat26 is clearly noted in Fig. 3b.

Fig. 3
figure 3

Numerical model validation and mesh sensitivity test: (a) Nusselt number at the stagnation point, (b) non-dimensional static pressure along the impinging plate.

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Turbulence model sensitivity test

This same validation of the Nusselt number at the stagnation point is presented in Fig. 4 by applying different turbulence models such as the standard k-ω, SST k-ω, standard k-ε, RNG k-ε and realizable k-ε model. Note that the standard k-ω model gives the best agreement with the experimental results26. In all simulations of the present study, we will adopt the mesh (2) containing 100,000 quadratic cells in addition to the standard k-ω turbulence model.

Fig. 4
figure 4

Nusselt number evolution at the stagnation point for different jet-plate distance: Turbulence model sensitivity test.

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Results and discussion

Evolution of the local Nusselt number

Figure 5 shows the evolution of the local Nusselt number (Nu) along the impinging plate for different jet numbers (N) and jet–jet spacing (S). The jet locations along the nozzle plate (Y = 0) are illustrated by the black circles for the different combinations of N and S. It is clear from this figure that increasing the number of jets (N) has no effect on the local Nusselt number Nu along the central zone (in the vicinity of the stagnation point X = 0). Beyond the axial location X = 5 and X = -5 (the lateral zone), increasing N result in higher Nu values indicating more intense local heat transfer exchanged between the flow and the impinging plate. On the other hand, increasing the jet–jet spacing (S), intensifies the local heat transfer along the central zone and weakens the same phenomena along the lateral zone.

Fig. 5
figure 5

Local Nusselt number evolution along the impinging wall for different numbers of jets and jet–jet spacing: (a) S = 1, (b) S = 2, (c) S = 3, and (d) S = 4.

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The lateral zone is spread over a wider length along the impinging plate compared to the central zone. That’s why it controls the global heat transfer between the flow and the impinging plate. Increasing N results in higher interaction between central and neighboring jets which intensifies the impact of the flow and consequently the local heat transfer along the lateral zone. This leads to improving the global heat exchanged along the impinging plate. On the other hand, higher S value results in lower interaction between central and neighboring jets. This results in a more direct and isolated impact of the central jet along the impinging plate, promoting local heat transfer along the central zone and weakens the same phenomenon along the lateral zone. As results, a lower global heat transfer exchanged is noted between the flow and the impinging plate. By increasing the jet–jet spacing S, jets interaction is reduced, leading to more independent cooling area and more uniform heat transfer distribution due to the weakening in the local Nusselt number value along the lateral zone. Lower Nu value along this zone results in a decrease in the overall heat transfer efficiency. While, by elevating the jets number (N), jets interaction increased leading to less uniform heat transfer due to the intensification of the Nu values along the lateral zone. Higher Nu value along this zone leads to an increase in the overall heat transfer along the impinging plate.

According to Fig. 5, along the central zone, an increase of 21.2% is noted for the maximum Nusselt number at the stagnation point when the spacing increases from S = 1 to S = 4 while no effect of varying the jets number is observed on this Nusselt number. On the other hand, along the lateral zone, when the number of jets increases from N = 5 to N = 11, an improvement in the maximum Nusselt number is observed by 24.5%, 21.2%, 15%, and 10.4% respectively for S = 1, 2, 3, and 4. This clearly shows that the increase in jet -jet spacing (S) slows down the enhancing effect of increasing the number of jets (N) on heat exchange along the lateral zone. Along the same lateral zone, we can also observe that decreasing (S) from 4 to 1 increases the maximum Nusselt number by 11.8%, 13.4%, 19.6%, and 26.1% when the number of jets (N) is equal to 5, 7, 9, and 11, respectively. This clearly means that increasing the number of jets (N) accelerates the enhancing effect of decreasing (S) on the heat transfer between the flow and the impinging plate. In summary, it is possible to improve the local heat transfer along the lateral zone and the global heat along the impinging plate by increasing the number of jets (N) and decreasing the jet–jet spacing (S). In addition, to improve the heat transfer along the central zone, it is better to increase jet–jet spacing.

Velocity magnitude contours

Figure 6 shows the velocity amplitude contours for different jets number (N) from 5 to 11 and different jet–jet spacing S = 1 (Fig. 6a), S = 2 (Fig. 6b), S = 3 (Fig. 6c) and S = 4 (Fig. 6d). The velocity amplitude is normalized by the initial velocity at the nozzle outlet. As clearly shown in Fig. 6, the velocity magnitude increases as the number of jets (N) increases. This is caused by the interaction between the central and neighboring jets to raise the velocity magnitude close to the impinging plate. This increases turbulence levels and, as a result, heat exchanged between the flow and the impinging plate. Jets interactions are more intense when increasing the number of jets. Fluctuations in velocity magnitude values may result from these interactions, with higher velocity magnitude zones occurring where jets connect. Greater jet–jet spacing (S) tends to isolate the effects of individual jets, producing more distinct velocity magnitude profiles and less interaction between jets. As a result of the decreased interaction, there may be a more uniform velocity distribution over the impinging plate, although the maximum velocity values may be lower. More intensive interaction between jets is obtained by smaller jet–jet spacing (lower S value), which could cause a localized increase in velocity magnitude at the zones where jets flow intersects. This is in good agreement with the results found by San et al.22 and Huber et al.28

Fig. 6
figure 6

Non-dimensional velocity magnitude contours for different numbers of jets and jet–jet spacing: (a) S = 1, (b) S = 2, (c) S = 3, and (d) S = 4.

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Because the convective heat transfer coefficient increases with increasing velocity magnitude and consequently turbulence level, heat transfer exchanged between the whole flow and the impinging plate is more intense. While a uniform velocity amplitude distribution can lead to more homogeneous heat transfer over the impinging plate. Note that large local differences in velocity magnitude can result in higher or lower heat transfer zones.

Streamlines contours

The contours of the streamlines and the axial velocity (along the y-axis) are plotted in Fig. 7 for a fixed jet–jet spacing S = 2 (Fig. 7a) varying the number of jets from 5 to 11. In Fig. 7b, the number of jets is fixed to N = 5 and the jet–jet spacing (S) varies from 2 to 4. The effect of varying the number of jets (N) on the vortical structure of impinging jet arrays can be deduced from Fig. 7a. The increase in the number of jets (N) multiplies the number of vortices in each recirculation zone between neighboring jets. Note that the circulation zone is characterized by an inverted flow with negative axial velocity (dark blue color). No effect of N is observed on the shape and size of vortices in each recirculation zone. Just the number of vortices is increased when elevating the number of jets (N). It seems also important to note that the only change in vortex size is noted for the two extremity vortices (out of the recirculation zones). These two vortices decrease in size when N is increased.

Fig. 7
figure 7

Streamlines and axial velocity contours: effect of varying the number of jets and jet–jet spacing: (a) S = 2, N = 5–11, (b) N = 5, S = 1–4.

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Figure 7b allows us to investigate the effect of the jet–jet spacing on the vortical structure of impinging jet array for constant value of the number of jets. As clearly shown in this figure, two counter-rotating vortices are present in the circulation zones between the jets close to the nozzles plate. The interaction and mutual attraction between neighboring jets is caused by these vortices. We can clearly observe that the vortices are similar in shape and size for small spacing value S = 1. In the recirculation zone (zone with inverted flow characterized by negative value of the axial velocity), it is even possible to say that every two vortices are symmetrical in each recirculation zone. However, we can clearly observe that the vortices’ shape changes as we increase the spacing to S = 2, 3, and 4. They are no longer equal in size, symmetrical, or similar. Beyond a jet–jet spacing value S = 1, when two neighboring jets get closer to one another, the pressure gradient widens the boundary layer, causing flow separation and the formation of larger, more complex, and noticeable vortices. Larger and more noticeable vortices may form if the jet–jet spacing (S) is increased toward S = 3 and S = 4. Streamlines can wrap around these vortices, forming distinct vortex patterns.

Static pressure contours

The dimensionless static pressure contours for different numbers of jets (N) and jet–jet spacing (S) are illustrated in Fig. 8. Pressure contours are plotted in Fig. 8a for N = 5 and for S = 1, 2, 3 and 4, the same contours are also showing for S = 2 and N = 5, 7, 9 and 11 in Fig. 8b. The nozzle locations along the nozzle plate (Y = 0) are illustrated by the black circles for the different combinations of N and S. As shown in Fig. 8b, when the number of jets (N) is increased at a fixed jet–jet spacing (S), the maximum static pressure on the impinging plate increases. This is explained by the expansion of a larger high-pressure zone due to adding more jets, which accentuates the flow impact along the impinging plate. In addition, more precise pressure peaks are observed with a higher number of jets (N), as each jet creates its own high-pressure zone around the point of impact. In contrast, Fig. 8a shows that increasing the jet–jet spacing (S) at a constant N value reduces the maximum static pressure along the impinging plate. The interaction between the spaced jets is less important, resulting in a more uniform pressure distribution and less pronounced pressure peaks.

Fig. 8
figure 8

Static pressure contours for different numbers of jets and jet–jet spacing: (a) N = 5 and S = 1–4, (b) S = 2 and N = 5–11.

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A clear correlation can be observed between the influence of the number of jets (N) and the jet–jet spacing (S) on the pressure and heat transfer: higher static pressure is associated with a more intense impact on the impinging plate and a thinner boundary layer, which favors better heat transfer. Thus, increasing N should lead to an increase in heat transfer, while increasing S should reduce it. Recirculation zones with negative pressure, where fluid flow is reversed, are clearly shown in Fig. 8, however, the areas of negative pressure are smaller than those of positive pressure, so they have no significant impact on the global heat transfer.

Turbulent kinetic energy and friction coefficient

In this section, we will attempt to explain the thermal behavior of the flow in the vicinity of the impinging plate by varying the jet–jet spacing (S) and the number of jets (N). This will be based on the evolution of the dimensionless turbulent kinetic energy (K) and the friction coefficient (Cf) along the impinging plate for different N and S values (Fig. 9). These curves are plotted for different number of jets N = 5, 7, 9 and 11 and for triple jet–jet spacing S = 1, S = 2 and S = 3. The different profiles shown in Fig. 9 show that, close to the stagnation point (central zone), there is almost no effect of the number of jets on the different values of turbulent kinetic energy and friction coefficient. This shows that the level of turbulence is unaffected by the increase of the number of jets in the vicinity of the central zone. However, as we move away from the stagnation point, we can clearly see an intensification of K and Cf by increasing N from 5 to 11. Which means no effect of varying N on the heat transfer along the central zone and an improvement effect of the same parameter outside of this region.

Fig. 9
figure 9

Evolution of the turbulent kinetic energy and the friction coefficient along the impinging plate for different numbers of jets, and jet–jet spacing: (a) K, S = 1, (b) K, S = 2, (c) K, S = 3, (d) Cf, S = 1, (e) Cf, S = 2, and (f) Cf, S = 3.

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Regarding the effect of jet–jet spacing, we can observe an intensification of the turbulent kinetic energy and friction coefficient values along the central zone as the jet–jet spacing (S) increases. While, beyond this zone, a weakening of K and Cf is clear as S increases. To conclude, the turbulence level remains insensitive to the jets number (N) in the vicinity of the stagnation point (the central zone). This level of turbulence in this same zone is improved by increasing the jet–jet spacing. Beyond the central zone, the turbulence level improves as the number of jets increases, while it weakens as the jets move further apart (increasing jet–jet spacing). With a higher number of jets (N), turbulence level as well as mutual interactions between jets tend to increase. This increasing turbulent agitation leads to more intense resistance to the flow, which is validated by the intensification of the friction coefficient along the impinging plate. Vortices and turbulent movements function as obstacles to fluid movement, increasing the force of friction. As a result, heat transfer is intensified by increasing the jet–jet spacing (S) in the stagnation zone and increasing the number of jets beyond this zone. This is confirmed by the Nusselt number profiles along the impinging plate previously shown in Fig. 5.

Mean Nusselt number

Along the present section, we will be examining the mean Nusselt number evolution as a function of three significant geometric parameters: the number of jets (N), the jet–jet spacing (S) and the jet-plate distance (H). A third parameter noted H appears along this section which represents the dimensionless distance between the nozzle plate and the impinging plate. Many researchers have found that the jet-plate distance has a considerable influence on the local and mean heat transfer in impinging jet array30,31,32,33,34,35,36,37. So, the mean Nusselt number has been expressed as a function of these three parameters through the development of a numerical correlation, where S, N, and H vary in the ranges of 1–4, 5–11, and 2–8, respectively. The numerical correlation is as follows:

$$\overline{{{\text{Nu}}}} = 67.41 \times {\text{H}}^{1.09} \times {\text{N}}^{0.41} \times {\text{S}}^{ - 0.14}$$
(22)

The evolution of the mean Nusselt number with the jet–jet spacing (S) for various jet numbers (N) and jet-plate distances of H = 2 (Fig. 10a), H = 4 (Fig. 10b), H = 6 (Fig. 10c), and H = 8 (Fig. 10d) is depicted in Fig. 10 along with the numerical correlation. Plotting symbols represent the numerical results based on simulations, and continuous lines represent the results based on the developed correlation. With a mean error of less than 2%, Fig. 10 amply demonstrates the satisfactory agreement between the developed correlation and the numerical results. This highlights the significance and accuracy of the established correlation as well as its capacity to estimate the mean Nusselt number concurrently and accurately as a function of the three parameters S, N, and H. As Fig. 10 makes abundantly evident, for all values of the jet-plate distance (H) and number of jets (N) that are taken into consideration, the mean Nusselt number decreases as the jet–jet spacing (S) increases. This suggests that the mean heat transfer between the fluid flow and the impingement plate is enhanced when the jet–jet spacing is reduced. As the number of jets (N) increases, the mean Nusselt number as a function of S decreases more sharply for the lowest jet-plate distance of H = 2 (Fig. 10a). Therefore, for lower jet-plate distances (H = 2), increasing the number of jets serves to amplify the effect of jet–jet spacing on the mean heat transfer. However, for the other H values (Fig. 10b–d), the number of jets (N) does not significantly influence the Nusselt number decrease as a function of S.

Fig. 10
figure 10

Mean Nusselt number variation as a function of the jet–jet spacing, jets number and jet-plate distance: (a) H = 2, (b) H = 4, (c) H = 6, and (d) H = 8.

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Higher means Nusselt number values are also obtained when the number of jets (N) is increased across all values of the jet-plate distance (H) and jet–jet spacing (S). This means that multiplying the number of jets improves the mean heat transfer exchanged in the present flow. When increasing H from 2 to 8 in Fig. 10a–d, a decrease in the mean Nusselt number is observed, independent of the N and S parameters. This supports the hypothesis that there is more convective heat transfer between the flow and the impinging plate when this impinging plate is positioned closer to the jet outlets.

Conclusion

In this study, numerical investigations were conducted on the dynamic and thermal characteristics of turbulent impinging jet array for a Reynolds number Re = 5500, a jet-plate distance H = 6 and considering various values of the jet–jet spacing (S) in the range [1–4] and number of jets (N) between 5 and 11. To predict the turbulent behavior of the present flow, standard k-ω turbulence model is adopted. Many important characteristics of the flow are presented for different values of (N) and (S) such as the local Nusselt number, velocity magnitude contours, streamline, axial velocity, static pressure contours. Evolution of turbulent kinetic energy as well as the friction coefficient is shown along the impinging plate. The global heat transfer between the flow and the impinging plate is investigated based on the average Nusselt number. Correlation, which predicts this latter parameter is performed as a function of N and S to which we added a third parameter (H) varied between 2 and 8. The results obtained in the present study can be summarized as follows:

  1. 1.

    The local Nusselt number evolution along the impinging plate shows that the best solution to intensify the local heat transfer along the lateral zone is to increase the jets number and decrease the jet–jet spacing. While, to improve the local heat transfer along the central zone, the only solution is to increase the jet–jet spacing (S), seeing that no influence of varying the number of jets (N) is observed along the same zone.

  2. 2.

    It is clear from velocity magnitude contours that, as the number of jets is raised, this velocity increases, mentioning higher interactions between the central and neighboring jets. More intense turbulence and heat exchange between the flow and the plate are the results of this increased interaction. Greater distance between individual jet isolates them, producing different velocity magnitude profiles and reduced jet interaction. Lower spacing values, however, increase the intensity of jet interaction and lead to localized increases in the magnitude of velocity where jets intersect which improved the convective heat transfer coefficient.

  3. 3.

    The vortical structure of the current flow is altered by changing the number of jets. The number of vortices in each recirculation zone between adjacent jets increases as the number of jets increases. Increasing the jets number, however, only affects the number of vortices; it has no effect on the size or form of the vortices within these zones. Notably, as N increases, the sizes of the two extremity vortices decrease. The vortices have comparable sizes and shapes for small spacing (S = 1). In fact, they seem symmetrical inside each recirculation zone. However, the vortices shapes change, becoming uneven, asymmetrical, and dissimilar as the spacing increases to S = 2, 3, and 4.

  4. 4.

    An increase in the number of jets at a constant jet–jet spacing raises the impinging plate’s maximum static pressure. The flow impact is emphasized by this wider high-pressure zone created by the additional jets. Maximum static pressure is decreased when jet spacing is increased while maintaining a constant number of jets. Better heat transfer is favored by higher static pressure, which increases impingement on the plate. Heat transfer should be improved by increasing the number of jets, but it might be decreased by increasing the jet–jet spacing. Negative pressure recirculation zones are evident, although they barely affect global heat transfer.

  5. 5.

    The number of jets has no effect on the turbulence level along the central zone. In this same zone, turbulence is improved by increasing the jet–jet spacing. Turbulence weakens with increasing jet–jet spacing and increases with a higher number of jets values outside of the central zone. Turbulence and the mutual interactions between jets intensify with higher values of the jet’s number. Higher flow resistance results from this increased turbulent agitation, as shown by the higher friction coefficient along the impinging plate. Frictional forces are increased by obstacles such as vortices and turbulent movements. As a result, adding more jets and increasing spacing along the lateral zone improve heat transfer along the stagnation zone.

  6. 6.

    Regardless of the jet-plate distance (H) and number of jets (N), the mean Nusselt number \(\overline{Nu}\) decreases as the jet–jet spacing (S) increases. This suggests that decreasing S improves heat transfer between the impingement plate and the fluid flow. As N increases, the impact of S on mean heat transfer is amplified for the lowest jet-plate distance (H = 2). The number of jets (N), however, has no discernible effect on \(\overline{Nu}\) decline with the S values for other H values. Increasing N over all H and S values results in higher \(\overline{Nu}\). This suggests that increasing N intensifies the average heat transfer. \(\overline{Nu}\) Consistently decreases from H = 2 to H = 8 (regardless of N and S values), corroborating with the idea that convective heat transfer is amplified when the impingement plate is nearer the jet outlets. \(\overline{Nu}\) is predicted through correlation and found to be in satisfactory agreement with the numerical values as a function of N, S, and H.