1 Introduction

In recent years, increasing the heat transfer coefficient due to its role in the economy and optimizing energy consumption in heat exchangers has been one of the topics of investigation. One of the methods of structure change in the turbulent boundary layer is to use a turbulator inside the boundary layer. For example, it is used to reduce the gas consumption in the heaters of natural gas pressure reduction stations. In this study, spherical perforated turbulators have been used to enhance the thermal performance of heat exchangers, reduce environmental impacts, and lower costs.

According to previous research, the present work focuses on the performance of the turbulator with aerodynamic shape and its effect on the characteristics of heat and flow. Bergles [1] conducted a comprehensive study on various methods to enhance thermal performance, categorizing them into thirteen heat transfer enhancement techniques, which can be further classified as passive and active methods.

Arasteh [2] studied the effect of the pitch spacing of rotational twist strips in a pipe, with a focus on analyzing thermal characteristics and flow behavior. The simulation of the effects of rotating torsion bars represented a novel addition to existing research. According to their findings, the Nu increased by 1.79 times compared to the smooth pipe. Ibrahim [3] researched the influence of a conical turbulator on thermal in a tube. According to previous studies, it was found that the effect of these conical rings is effective in increasing heat transfer. This simulation was done to complete previous studies. The findings of the study in the range of Re = 25000–6000 show that the highest thermal performance is 1.291.

Sharma [4] investigated the influence of a perforated hollow oval turbulator in a tubular heat exchanger on heat transfer. The results show that in the range of Re = 10000–35000, an increase in heat transfer and thermal performance coefficient was reported from 164% to 281% and 111% to 130%.

Patil [5] researched the heat properties in a pipe with hexagonal annular turbulators. In this study, a hexagonal ring turbulator is proposed. The working fluid of air is in the Re = 6000–24000. The Nu is 3, f increases by 3.30 folds compared to the smooth pipe, and \(\eta \) =1.34.

Abdelmaksoud [6] investigated the effect of using a vortex generator in the heat exchanger pipe to improve thermal performance. This study was done to complete the previous studies with three straight configurations, two curves, and three curves. The working fluid of water is in the Re = 10000–2000. According to the results, the highest \(\eta \) is 1.24.

Rajan [7], evaluated the performance of the pipe with a double-cut twisted strip insert. The results for the Nusselt number, research findings showed that the thermal performance increased by 0.33–11.63% compared to the plain pipe.

Liu [8] numerically analyzed the optimization and exergy of nanofluid inside the pipe with a spiral turbulator. The results showed that with the increase of Re, the fluid velocity increases, and exergy losses decrease. In addition, the temperature decreases with increasing flow turbulence.

Padmanabhan [9], studied the characteristics of heat transfer in a heat exchanger with a spiral insert using computational fluid dynamics. In this study, an attempt was made to increase the heat transfer with a reasonable pressure drop.

Manh [10], analyzed the effect of twisted tape in a heat exchanger with nanofluid on exergy and heat transfer. The numerical simulation with the k-ε turbulence model (RNG) was used. The results showed that the amount of exergy decreases by more than 35% with the increase of the height ratio in a certain range.

Suri [11] studied the influence of square holes created in multiple twist strips on the thermal specifications of a pipe. They found that using multiple twisted strips increases the Nu and f of the heat exchanger by 4.45 and 70 times, respectively.

Alizadeh [12], investigated the effect of turbulator geometry on heat transfer in a channel with a square cross-section. The innovation of the work is the simultaneous use of turbulator to increase agitation, more heat transfer and increase the contact surface, for this purpose, different turbulators were used.

Bartwal [13] investigated the heat performance and flow characteristics of a monotonic heat flux using wire grids in a pipe. The result indicates that wire mesh significantly affects Nu and f in the Re = of 5000 to 40000.

Kumar [14] evaluated the heat exchanger with different geometries of twisted turbulators using a ternary nano-hybrid. In this study, the effect of turboblators and nanofluid on increasing heat transfer and reducing emission was investigated. The results of each method were reported in terms of cost and technical analysis. Azizi [15] investigated the influence of a twisted band turbulator in a pipe on the flow characteristics. In this study, the working fluid was air-water. The highest pressure drop and Nusselt number increase were reported as 81% and 35%, respectively.

Khanmohammadi [16], using the laws of thermodynamics, evaluated a dimpled pipe with a conical turbulator. In this study, the effect of pitch and heat flux on Nusselt number, entropy production, and pressure drop in the Reynolds number range of Re = 5000–20000 was investigated. In this study, the effect of pitch and heat flux on Nusselt number, entropy production, and pressure drop in the Reynolds number range of Re = 5000–20000 was investigated. The results showed that the indentation on the pipe body, the appropriate choice of step, and the number of indentations, can be a suitable method to increase heat transfer. Mousavi and Alavi [17] studied the influence of the airfoil to optimize the flow and thermal in the heat exchanger. The working fluid of air and water is in the Re = 6000–10000. In this study, the Nu increased by 2.6 fold. The highest \(\eta \) is 1.91.

Firat [18] experimentally investigated the performance and entropy production of angular blade inserts as a heat transfer enhancer. The results showed that the turbulator with blade angles of 60 degrees is more advantageous than the rest of the studied geometries due to less entropy production. Nalavade [19] researched the influence of flow divider-type turbulators inside the pipe to analyze the turbulent fluid flow and heat performance. The air as a working fluid is used in the Re = 7000–21000. The Nu and f changed by 1.6 and 5.71, respectively. The highest \(\eta \) is 1.43.

Banihashemi [20] investigated the effect of the rotation of a disc turbulator on heat changes in a pipe. The heat transfer coefficient increased by 62% compared to a smooth pipe.

Banihashemi [21, 22] studied the effects of 90-degree and complete rings on performance in a pipe. The rotational ring,\(\eta \), and Nu increased in the range of 0.88–2.1 and 2.6–4, respectively.

Pan [23], Using perforated magnetic turbulators, investigated the heat transfer enhancement in a heat exchanger experimentally. The results showed that simple and perforated turbulators increased heat transfer by 156% and 150%, respectively. In addition, the maximum thermal performance was reported as 2.06.

Ravanbakhsh [24] studied the thermal and flow characteristics in a domestic gas water heater with different geometries for the pipe with turbulator. According to the two parameters of pressure drop and Nusselt number, they determined the optimal Reynolds number for cold flow using the multi-objective optimization method. The studies related to the types of turbulators in the pipe are summarized in table 1.

Table 1 Research results related to the use of turbulators in pipe.
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So far, limited studies have been conducted regarding the use of spherical turbulators inside pipes. Investigating the turbulator with an aerodynamic shape and its impact on the thermal and turbulent flow characteristics is a new topic in industrial applications [35,36,37,38,39,40]. It is important in reducing energy consumption, increasing efficiency, and heat transfer enhancement. So this study investigates the performance of spherical turbulators with different geometries in the pipe with a working fluid of air under the influence of uniform heat flux. "Six spherical turbulators with elliptical, circular, square, and hexagonal perforations with equal obstruction surfaces are placed inside the pipe."

Based on the research done, the innovation of this article be expressed as follows:

  • A limited of studies have been conducted on the use of aerodynamic turbulators in in-pipe airflow

  • The investigation of performance in the pipe with turbulators with the same blocking surface inside a pipe has not been investigated in studies so far.

  • The effect of the geometry of the square, circle, hexagon, and oval holes on the spherical turbulator has not been investigated in previous studies.

2 Problem description and solution methodology

2.1 Turbulator geometry

Six perforated spherical turbulators with a 40% obstruction surface have been designed inside the pipe using the design model environment. Figure 1 shows the pipe geometry with used turbulators and related details. Also, Table 2 shows the dimensions considered for the tube and Korean turbulators in the present work.

Figure 1
figure 1

Pipe geometry with used turbulators and related details.

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Table 2 Parameters and dimensions considered for the model in the present work.
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2.2 Mesh generation

The multi-zone method is used in smooth pipe mesh. By doing this method, the networking environment is connected to the (ICEM) environment. With this work, it is possible to produce an organized mesh. Also, boundary layer meshing is used for more accurate results. Figure 2 shows the smooth pipe mesh. The O-grid method is used in this mesh. This method is usually used for circular cross-sections

Figure 2
figure 2

The meshing of the smooth pipe from the lateral and opposite sides.

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Independence from the mesh is done in all cases of numerical analysis considered in this study, and the value of the Nusselt number is considered an indicator for choosing the best mesh. Table 3 shows the information about the four grids of the smooth pipe at a Re = 10000. Based on this, the Nusselt number in the number of 514556 elements has few changes compared to before.

Table 3 The Nu in four different meshes of the smooth pipe at Re = 10000.
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The tetrahedrons method has been used to mesh the pipe in the presence of perforate spherical turbulators. In figure 3, the grid generation is shown.

Figure 3
figure 3

Pipe meshing with turbulators.

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2.3 Governing equations

To obtain the governing equations and solve the problem, the following assumptions are considered:

  1. 1.

    The flow is steady and incompressible.

  2. 2.

    Radiation heat transfer is ignored.

  3. 3.

    The effect of gravity is ignored.

The basic structure of problems related to fluid mechanics and heat transfer are equations that are derived directly from the laws of conservation of fluid physical properties.

Equation of continuity:

$$ \frac{\partial \rho }{{\partial t}} + \frac{{\partial (\rho u_{i} )}}{{\partial x_{i} }} = 0 $$
(1)

Equation of momentum:

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

Equation of energy:

$$ \frac{\partial }{\partial t}\left( {\rho E} \right) + \frac{1}{2}\frac{\partial }{{\partial x_{i} }}\left[ {\mu_{i} \left( {\rho E + \rho } \right)} \right] = \frac{\partial }{{\partial x_{i} }}\left[ {k_{eff} \frac{\partial T}{{\partial x_{i} }}} \right] $$
(3)

The equations of the k- ω -SST model are summarized as follows [41]:

Equation of turbulent kinetic energy [42]:

$$ \frac{\partial (\rho k)}{{\partial t}} + \frac{{\partial (\rho u_{i} k)}}{{\partial x_{i} }} = \tilde{P}_{K} - \beta^{*} \rho \omega k + \frac{\partial }{{\partial x_{i} }}\left[ {(\mu + \sigma_{k} \mu_{t} )\frac{\partial k}{{\partial x_{i} }}} \right] $$
(4)

Specific loss rate equation:

$$ \begin{aligned} &\frac{\partial (\rho \omega )}{{\partial t}} + \frac{{\partial (\rho u_{i} \omega )}}{{\partial x_{i} }} = \frac{\upalpha }{{\upnu _{t} }}\tilde{P}_{K} - \beta^{*} \rho \omega^{2} + \frac{\partial }{{\partial x_{i} }}\left[ {(\mu + \sigma_{k} \mu_{t} )\frac{\partial \omega }{{\partial x_{i} }}} \right] + 2\rho (1 - F_{1} )\sigma_{{\omega^{2} }} \frac{1}{\omega }\frac{\partial k}{{\partial x_{i} }}\frac{\partial \omega }{{\partial x_{i} }} \hfill \\ & \tilde{P}_{K} = \min \left[ {\mu_{t} \frac{{\partial u_{i} }}{{\partial x_{j} }}\left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right),10\beta^{*} \rho k\omega } \right],\;\upnu _{t} = \frac{{a_{1} k}}{{\max (a_{1} \omega ,SF_{2} )}} \hfill \\ & F_{1} = \tanh \left. {\left. {\left\{ {\left\{ {\min \left[ {\max \left( {\frac{\sqrt k }{{\beta^{*} \omega y^{\prime}}}\frac{{500\upnu }}{{y^{2} \omega }}} \right),\frac{{4\rho \sigma_{{\omega^{2} }} k}}{{CD_{k\omega } y^{2} }}} \right]} \right.} \right.} \right\}^{4} } \right\},\;F_{2} = \tanh \left[ {\left[ {\max \left( {\frac{2\sqrt k }{{\beta^{*} \omega y^{\prime}}}\frac{{500\upnu }}{{y^{2} \omega }}} \right)} \right]^{2} } \right] \hfill \\ & \varphi = F_{1} \varphi_{1} + (1 - F)\varphi_{2} ,\;CD_{k\omega } = \max (2\rho \sigma_{{\omega^{2} }} \frac{1}{\omega }\frac{\partial k}{{\partial x_{i} }}\frac{\partial \omega }{{\partial x_{i}^{\prime } }}10^{ - 10} ) \hfill \\ \end{aligned} $$
(5)

All the constants are a combination of constants related to the k-ε model, and k-omega are calculated. The constants for this model are[41, 43]:

$$ \beta^{*} = 0.09,\varphi_{1} = 5/9,\beta_{1} = 3/40,\sigma_{k1} = 0.85,\sigma_{\omega 1} = 0.5,\varphi_{2} = 0.44,\beta_{2} = 0.0828,\sigma_{k2} = 1,\sigma_{\omega 2} = 0.856 $$

2.4 Boundary conditions

The boundary condition at the inlet of the smooth pipe is the flow velocity in the range of Re = 6000–24000. By solving several times and obtaining the developed flow velocity profile, the inlet velocity for the numerical solution is considered as the outlet velocity in the smooth pipe. The air temperature at the pipe inlet is 300 K. For turbulence, the hydraulic diameter is 62 mm and the intensity of turbulence is 5%. Turbulence intensity is the root ratio of the average speed disturbances to the average flow velocity. At the outlet of the pipe, the relative pressure is zero. A heat flux of 1000 W/m2 is considered for the wall boundary condition. In a computational fluid dynamics (CFD) simulation, an outlet pressure of zero typically means that the boundary condition at the outlet of the domain is set to atmospheric pressure. This assumes that the flow exiting the domain is unrestricted and is discharging into the surrounding atmosphere.

2.5 Numerical solution

Ansys Fluent software was used for the numerical study. In this work, equations are solved using the basic pressure solver based on the type of flow. Quadratic discretization is used for pressure interpolation. The second-order upwind method is used for the turbulent kinetic energy equation and the Quick algorithm for momentum and energy equations and the coupling algorithm is used simultaneously to solve the speed and pressure profile. The energy equation is also dissolved due to the temperature changes and the heat transfer of the wall. The convergence criteria are considered less than 10−3, 10−3, and 10−6, for continuity, momentum, and energy equations.

2.6 Turbulence model

In the numerical method, the K- ω -SST model was employed to model turbulent flow. Jasinski [44] used the k-w: SST model to study the effects of spherical turbulators inside the heat exchanger pipe. Table 3 shows the comparison of the value of the Nusselt number in the pipe with a 45-degree disc turbulator at a speed of 300rpm at Re = 6000 by different models with experimental results. The maximum difference between the results predicted by the k-ε RNG model and the experimental results is in the range of 5.9%, so this model has good accuracy. In addition, for predicting rotating flows the RNG model is very suitable.

2.7 Calculation formulas

The parameters of the Nu, Re, and f are expressed as follows:

$$ Nu = \frac{{h_{i} D}}{k} = \frac{{q_{w} d_{t} }}{{k(T_{w} - T_{b} )}} $$
(6)
$$ {\text{Re}} = \frac{\rho VD}{\mu } $$
(7)
$$ f = \frac{\Delta P}{{\frac{1}{2}\frac{L}{D}\rho V^{2} }} $$
(8)

The effectiveness of a heat transfer enhancement technique is evaluated by the Thermal Performance Factor which is a ratio of the change in the heat transfer rate to the change in friction factor. The coefficient of thermal performance is calculated from the following equation to check the justification of the flow stimulation with the turbulator and the effect on the optimization of energy consumption [45].

$$ \eta = \left(\frac{Nu}{{Nu_{s} }}\right)\left(\frac{{f_{s} }}{f}\right)^{\frac{1}{3}} $$
(9)

2.8 Independence from Mesh

Figure 4 illustrates the Nusselt number and friction factor variations for four different grid configurations at Re = 10,000. Also, in table 4, the number of selected elements is displayed according to the changes in Nusselt number in different meshes.

Figure 4
figure 4

Nusselt number changes with the node number.

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Table 4 Nusselt number and friction factor change in the pipe with different inserts.
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2.9 Validation of numerical results

Figures 7 and 8 show the value of the Nu and f in this research with analytical equations. Dittus and Bolter [46] expressed the 10 equation for the Nu:

$$ Nu = 0.023{\text{Re}}^{0.8} \Pr^{0.4} $$
(10)

Gnielinski [47] expressed the 11 equation for the Nu:

$$ Nu = \frac{{(\frac{f}{8})({\text{Re}} - 1000)\Pr }}{{1 + 12.7(\frac{f}{8})^{\frac{1}{2}} (\Pr^{\frac{2}{3}} - 1)}},3 \times 10^{3} \le {\text{Re}} \le 5 \times 10^{5} $$
(11)

Petukhov [48] expressed the 12 equation for the Nu:

$$ Nu = \frac{{(\frac{f}{8}){\text{Re}} pr}}{{1 + 12.7(\frac{f}{8})^{\frac{1}{2}} (\Pr^{\frac{2}{3}} - 1)}},10^{4} < {\text{Re}}_{{}} < 5 \times 10^{5} \;,\;0/5 < {\text{Pr}} < 200 $$
(12)

Moody's equation [49] expressed the 13 equation for the f:

$$ f = 0.316\;{\text{Re}}^{ - 0.25} ,\;{\text{Re}} \le 20000 $$
(13)

Petukhov [48] expressed the 14 equation for the f:

$$ f = (0.79\ln {\text{Re}} - 1.64)^{ - 2} \;,3 \times 10^{3} \le {\text{Re}} \le 5 \times 10^{6} $$
(14)

According to figure 5, the amount of error depends on the Reynolds number, and for the Nusselt number, this amount of error is recorded compared to as Dittus and Bolter equation 12%, Gnielinski is 8%, and Petukhov is 6% respectively. Also, according to figure 6, for the friction coefficient, the error rate compared to Petukhov's equation is 8.1%, Moody's is 4.2%, and Blasius's is 5.9%.

Figure 5
figure 5

Value of the Nu for the plain pipe.

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Figure 6
figure 6

Value of f for the plain pipe.

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To confirm the results of the numerical solution of this article, the geometric model of the pipe with a solid sphere was examined. This model is derived from the work of Jasiński et al [44]. In similar Reynolds numbers, the value of the friction coefficient obtained from the numerical solution was compared with the experimental work. As shown in figure 7, the maximum deviation between the stated results is less than 8%.

Figure 7
figure 7

Numerical simulation valuation results.

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3 Results and discussion

Before examining the effect of spherical turbulators with different perforations on heat transfer parameters, let's first compare the pressure drop, heat transfer coefficient, and temperature increase in a pipe with a single turbulator and multiple turbulators. According to table 5, the pressure drop of the fluid and the values of the heat transfer coefficient in the case of multiple spherical turbulators are higher compared to a single turbulator. By calculating the average of the pressure drop values in the case of multiple turbulators and a single turbulator, the pressure drop increases by almost 6.2 times compared to the smooth pipe. Additionally, by averaging the heat transfer coefficient values, it can be observed that the heat transfer coefficient in the case of multiple turbulators increases by 32% compared to the single turbulator case. In the presence of the turbulator in the pipe, the energy of the particles in the collision with the turbulator and other particles increases, and as they move away from the stationary state, the thickness of the boundary layer decreases. In the case where there are several turbulators in the pipe, this issue is repeated alternately.

Table 5 Comparison of fluid pressure drop and heat transfer coefficient in two cases: a pipe with one and several perforated spherical turbulators at Re = 10000.
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As the number of turbulators increases, the amount of energy loss increases due to the production of more vortices. In addition, with the increase in the number of turbulators, the friction coefficient increases due to the increase in the contact surface of the turbulator with the air. Also, with the increase of turbulators, the Nusselt number increases due to the increase in flow turbulence, the production of more vortices, and more mixing of the fluid flow next to the wall. It should be noted that the number and intensity of these vortices have a significant effect on the Nusselt number according to the geometry used. According to figure 8, because the square hole produces larger vortices closer to the pipe wall, it has a little more effect on the Nusselt number. Also, due to the increase in the axial velocity of the flow, the Nusselt number is higher than other geometries. Because the formation of the vortex increases the amount of hydraulic energy loss of the fluid, therefore, this issue increases the amount of fluid pressure loss. According to figure 10, it can be seen that the change in the speed profile leads to a change in the temperature profile. In figure 8, the temperature contour is drawn on a central plane along the length. In this figure, the heat penetration in the place of the turbulators is clear.

Figure 8
figure 8

Temperature contour in pipe with spherical turbulators with various perforations at Re = 10000.

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Also, table 6 shows a summary of the results in pipe with several spherical turbulators at Re = 25000–60000. In the continuation of the article, the results are explained separately.

Table 6 Summary of the results of the present study at Re = 25000–60000.
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3.1 Investigation of thermal characteristics

Figure 9 illustrates the temperature distribution along the length of the pipe in the presence inserts. The obstruction created by the turbulators, leading to strong secondary flow near the pipe walls, results in thinning of the thermal boundary layer, which is one of the physical reasons for enhanced heat transfer. The temperature in the region between the turbulators and the pipe wall is higher due to the constant heat flux from the pipe wall and the formation of fluid jets near the tube walls compared to the central region of the pipe. The flow disturbance obstacles near the pipe walls increase and create recirculation of the flow between the central area and the pipe walls. This means that flow mixing significantly increases in the presence of perforated spherical turbulators, leading to increased turbulence intensity in the fluid flow through the heat exchanger pipe. The disruption of the thermal boundary layer due to the rotational flow occurring through the holes in the turbulators contributes to raising the fluid temperature. Due to the differences in geometry between the hexagonal and circular perforations and the obstruction created by the turbulators, the temperature near the wall is more severe in pipes with circular perforated turbulators compared to those with hexagonal perforated turbulators.

Figure 9
figure 9

Temperature contour in pipe with spherical turbulators with various perforations at Re = 10000.

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3.2 Investigation of velocity

Figure 10 shows the velocity contour along the pipe for different geometries. In figure 10(a), due to the presence of the turbulator, the flow area is reduced., as a result, the fluid velocity increases while passing through the holes. This speed increase causes turbulence in the flow and increases the heat transfer coefficient. In figure 10(b), due to the type of hexagonal geometry, the flow in the stage of passing through the turbulators has more gaps than other geometries, which causes a more robust secondary flow phenomenon that can occur and ultimately leads to a thinning of the thermal boundary layer. In figure 10(c), unlike the hexagonal and circular geometries, the impingement flow has less with the area between the turbulators and the pipe wall, resulting in high heat transfer. Figure 10(b) presents the distribution of velocity along the pipe with hexagonal obstructions. Consequently, fluid velocity increases as it passes through the holes. The hexagonal geometry introduces interruptions in the flow path compared to a circular hole. In the case of a circular hole, secondary flow phenomena behind the turbulators are more pronounced, resulting in a more substantial separation of the flow in that region and consequently exerting a stronger shear force on the pipe wall. As it progresses along the pipe, turbulence and flow velocity increase, and denser vortices are formed. It leads to heat transfer increases. The vortices formed in the pipe with turbulators with a square hole have a more irregular shape than the hexagonal geometry, which can be due to the type of square geometry. As evident in figure 10, due to the continuous collision of the fluid flow with the turbulators, the regular structure of the flow is disrupted, and more fluid along the pipe as it progresses, and the formed vortices become denser.

Figure 10
figure 10

Velocity distribution in the pipe with turbulators at Re = 10000, (a) hexagonal (b) circular (c) oval (d) square.

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Figure 11 shows the flow lines in the pipe with turbulators under study. The flow velocity at the center of the pipe is the maximum value. When the turbulators are placed in the flow path, the flow has a regular structure at first, but after hitting the first hollow spherical turbulator, the fluid flow becomes turbulent and irregular. In this case, the increase in thickness of the boundary layer does not continue. In figure 11(b) due to the type of hexagonal geometry, the flow in the stage of crossing the boundary can cause a stronger secondary flow to occur and ultimately lead to the thinning of the boundary layer and an increase in heat transfer. In figure 11(c), the fluid flow passes through the six square holes and flows, the flow has relatively less contact with the area between the pipe wall and as a result, the heat transfer is better than the geometry of others. In figure 11(d) is observed that the flow lines increase compared to other geometries. This high speed causes the flow to be confused more and increases the heat transfer compared to the geometries.

Figure 11
figure 11

Velocity continuous distribution in the pipe in the presence of turbulators with (a) circular (b) hexagonal (c) square (d) oval holes at Reynolds number 10000.

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In figure 12, the fluid jet next to the pipe wall affects the thermal boundary layer, and as a result heat transfer increases. The hot flow is directed towards the center of the pipe at a faster speed and the mixing of the flow increases. Figure 12 shows that behind the turbulator, the vortex is formed, which is larger and wider for the oval hole. For the hexagonal hole, a vortex has been created, which causes less pressure drop.

Figure 12
figure 12

Flow line in a pipe with turbulators with (a) Square, (b) Oval, (c) Circular, (d) Hexagonal hole at Re = 10000.

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3.3 Investigation of Nusselt number

Figures 13 and 14 depict the dependence of the Nu and Nu/Nus with the Re in a pipe in the presence of perforated spherical. At high Reynolds, the flow turbulence is greater. This increase in turbulence affects the structure of the thermal boundary layer. In addition, due to better mixing and better flow, the Nusselt number increases. The turbulator with an oval hole produces more vortices closer to the pipe wall, so it is more effective than other geometries. The Nu/Nus decreases with increasing Re. For turbulators, vortices of different sizes are formed behind the turbulators, because the location of these vortices is at a different distance from the place of heat transfer, i.e. the pipe wall, the Nusselt number is different for each of the turbulators. In the pipe equipped with a circular hole turbulator Nu increases by 146%-97% compared to the smooth pipe. For the pipe with a hexagonal turbulator, the highest Nusselt number is 38.95 at Re = 25000. The maximum Nusselt number for the pipe with the spherical turbulator of the oval hole is 149.03 at the Re = 25000. The Nu for square holes increases by 58–89%. Also, the Nu/Nus for hexagonal holes increases between 124–165% compared to the smooth pipe. In general, with the used turbulators, the Nusselt number becomes 2.81.

Figure 13
figure 13

Nu v.s. Re.

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Figure 14
figure 14

Nu/Nus v.s. Re.

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3.4 Investigation of Friction factor

Figure 15 depicts the pressure drop in the pipe. As the Reynolds number increases, the value of the friction coefficient decreases, and the reason for this is the predominance of the effects of friction compared to inertia, and the increasing effect of the vortices formed behind the turbulators. Any geometry that has good aerodynamics, its a friction coefficient is lower than others. The pressure drop increases due to the presence of the turbulator and the reduction of the flow passing surface, as well as the increase in the air velocity. As seen in figure 17, the higher the Re, the more turbulence of the flow increases due to the increase in the flow velocity. The pressure drop caused by the placement of perforated spherical turbulators increases compared to the smooth pipe with the increase of the Re. This increase was due to the blockage created by the turbulators in the flow path and creating a local drop. In turbulators with hexagonal holes, the pressure drop caused by the drag force on the inserts is less, which causes a lower pressure drop, as a result, the pressure drop increases less than other geometries. In other words, according to equation 8, if the friction coefficient is assumed constant, the pressure drop is proportional to the second power of the speed. The pressure drop in the pipe does not depend on the heat flux entering the pipe walls, but only on the Reynolds number. The reason for the lower pressure drop in the pipe with spherical turbulators with hexagonal holes compared to other geometries can be explained by the fact that in turbulators with hexagonal holes, due to the type of geometry, there is less obstruction in the flow path, which causes less local drop and finally drop The pressure increases less than other geometries. As can be seen, the pressure drop increases as the fluid flow progresses along the pipe due to the presence of turbulators and the increase in fluid flow velocity. Among the studied geometries, the oval hole creates the highest pressure drop due to the creation of more local drops than other geometries in the pipe with spherical turbulators. Among the turbulators under investigation, the highest and the lowest pressure drops are related to spheres with hexagonal and oval holes, respectively.

Figure 15
figure 15

Variations of pressure drop in pipes with turbulators.

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The friction coefficients increase due to the introduction of turbulators in the flow path, leading to the generation of vortices near the wall and the turbulators. According to figure 16, the most significant increase in the friction coefficient is observed in the pipe with spherical turbulators equipped with a square hole at Reynolds 15,000, with a 16.24-fold increase. The most substantial increase in the friction coefficient for the hexagonal turbulator is observed at Re = 6,000, with a 3.77-fold. The most significant increase in the friction coefficient with spherical turbulators featuring an oval hole occurs at Re = 25,000, increasing by 13.66 times compared to a smooth pipe.

Figure 16
figure 16

Variations f/fs with Re in a pipe with turbulators.

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3.5 Investigation of thermal performance

Figure 17 shows the variations Nu/Nus with f/fs in the pipe equipped with perforated spherical turbulators with different geometries. The highest Nu/ Nus and f/fs can be seen in the pipe with spherical turbulators with oval and hexagonal holes. In the investigations, it can be noticed that the difference in the ratio of the Nusselt number (Nu/ Nus=1.58–2.8) is very low, while the ratio of the friction number is very large (f/fs= 46.3–46.24).

Figure 17
figure 17

Nu/Nus v.s with f/fs.

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Figure 18 shows that thermal performance decreases with Reynolds number. The reason for the reduction of the thermal performance with the Reynolds number is the formation of the turbulence intensity on the flow excitation. According to figure 18, the range of thermal performance coefficient in the tube with spherical turbulators with square, oval, hexagonal, and circular holes is obtained in the range of 0.98–1.20, 1.03–1.25, 1.02–1.21 and 0.9–1.18, respectively.

Figure 18
figure 18

η v.s Re.

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3.6 Investigation of Thermal performance of pipe with an oval hole in the stretched state in the perpendicular direction

According to the results obtained in the previous parts, to complete the study, oval holes in the stretched state in the vertical direction were also examined. Figure 19 illustrates the spherical turbulator with oval perforation in the vertical direction.

Figure 19
figure 19

Schematic of a stretched spherical turbulator with oval holes in the vertical direction.

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A change in the velocity profile results in a change in the temperature profile. In figure 20, the temperature contour is drawn on a plate in the center and along the length of the pipe. In this figure, the penetration of heat at the place of obstacles is clear. Also, when the fluid hits the turbulator due to high agitation and agitation, the temperature of the fluid starts to increase from the side of the wall.

Figure 20
figure 20

Temperature and velocity contours of a tube with oval vertical hole turbulators.

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Figure 21, depicts variations in the coefficient of thermal performance as a function of the Reynolds number, the highest η is achieved 1.16, which is lower compared to the other geometries studied.

Figure 21
figure 21

η v.s. Re in a pipe with turbulators with oval perforations stretched in the vertical direction.

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4 Conclusion

Enhancing heat transfer is a crucial concern in energy management, significantly impacting cost reduction and economic benefits. Researchers are keenly interested in improving heat transfer in heat exchangers through the use of various barriers. The findings of these studies hold promise for industrial applications, aimed at boosting efficiency. This research focuses on examining the effect of perforated spherical inserts with varying geometries on thermal efficiency and flow structure through numerical analysis. The results are:

  1. 1.

    The pressure drop in the pipe is much lower with spherical turbulators with hexagonal holes than with other geometries.

  2. 2.

    In the pipe with turbulators with an oval hole, the highest Nusselt number increase is obtained, 2.81 times more than the smooth pipe.

  3. 3.

    In the pipe equipped with spherical turbulators with an oval hole, the η has increased by 1.25 times. This represents the highest improvement among all studied geometries.

  4. 4.

    The range of Nusselt number changes for models with circular, hexagonal, square, and oval holes increases by 1.97–2.46, 2.24–2.65, 1.58–1.89, and 2.48–2.81 times, respectively.

  5. 5.

    A perforated spherical turbulator with a hexagonal hole creates less pressure drop than other geometries, and also, the highest performance coefficient with this geometry is 1.21.

  6. 6.

    Aerodynamic turbulators have a positive effect on thermal performance.