Introduction

Heat exchangers play a vital role in facilitating energy transfer for large-scale equipment or compact electronic devices. The main challenge in designing the heat exchanger system is achieving enhanced heat transfer efficiency while minimizing the pumping power requirements, particularly within limited space constraints. To address this, the use of vortex generators (VGs) as a passive heat transfer enhancement technique has been broadly investigated in various industrial applications. VGs exhibit the potential to improve heat transfer through several mechanisms. These mechanisms include secondary vortices or swirls, mixing the flow along the wall of the main flow, reducing the thickness of the thermal boundary layer, and increasing turbulence intensity [1]. These mechanisms highlight the promising capabilities of VGs in enhancing overall heat transfer efficiency in diverse engineering applications.

Circular pipes have widespread applications in thermal power plants, chemical process plants, and solar heating. Ajarostaghi et al. [2] studied an innovative VG with eighteen blades and MWCNT-Fe3O4/water nanofluid in a circular pipe. They found that a higher heat transfer rate was achieved by using both VG and hybrid nanofluid techniques. Silva et al. [3] examined two types of VGs, namely, delta-winglet and rectangular-winglet VG, within a circular pipe for different angles of attack (α) (15°, 30°, and 45°). The findings demonstrated that both VGs at an α of 45° achieved the highest heat transfer, while the delta-winglet VG with an α of 30° exhibited the best thermo-hydraulic performance. Yang et al. [4] employed three longitudinal VGs in a vertical tube with an upward flow of supercritical CO2. They found that using 24-row longitudinal VGs resulted in a remarkable 73.4% enhancement in the thermal efficiency index. However, the heat transfer deteriorated when the vortices only covered a portion of the tube wall. Zhai et al. [5] explored the thermal performance of delta winglet VG pairs in a circular pipe with different pitch ratios, flow directions, and configurations. It was found that the best thermal enhancement factor was achieved through the combination of winglet configuration with common flow-down, downstream flow direction, and optimized winglet designs. In addition, the pitch ratio was crucial in affecting the Nusselt number and friction factor.

Rectangular or mini-channel heat sinks have been developed for cooling electrical chips and electronic components. Most studies have focused on parameter optimizations [6,7,8] and proposed new configurations [9,10,11] of longitudinal VGs. Brodnianska et al. [12] investigated the thermal performance of cylindrical VGs in a wavy rectangular channel. The results showed that the Nusselt number increased with the reduced channel height and rising Reynolds number. Moreover, the combination of cylindrical VGs and wavy channels caused the Nusselt number to be 6.28 times higher than a smooth pipe. Datta et al. [13] examined the heat transfer behavior of longitudinal VG pairs with various inclinations and positions in a rectangular micro heat sink. The best thermal performance could be obtained by combing two pairs of longitudinal VGs with an α of 30° when the Reynold number exceeded 600. Additionally, a longer channel downstream of the second VG pair caused an increase in heat transfer due to better fluid mixing following the vortex breakup. Karkaba et al. [14] investigated VGs with different longitudinal pitches in a rectangular channel. They observed that the thermal enhancement factor could be increased by 90% by using five rows of VGs at a longitudinal pitch equal to three times the channel height. Demirag et al. [15] reported the heat transfer augmentation for the conic VG with different α, blade angles, and scale ratios in a rectangular solar air heater. The greatest thermal enhancement factor of 1.316 was obtained with an α of 37.5°, blade angle of 30°, and scale ratio of 1:1. Fu et al. [16] studied the longitudinal VGs with different configurations in a mini channel with varying heights. They found that the mixed arrangement with the one-sided common flow down and another-sided common flow up at a channel height of 1 mm performed the best heat transfer, with a Nusselt number enhancement of 28–35% when compared to the smooth channel. Kashyap et al. [17] explored the effect of surface shape modification of the rectangular VG on heat transfer in a rectangular plate. They concluded that multiple concave textures in the leading face and a single convex texture in the trailing face significantly strengthened the primary vortex, leading to enhanced heat transfer. Ke et al. [18] conducted a study on the heat transfer performance of the delta-shaped winglet with combined common-flow-down and common-flow-up configurations in a rectangular channel, where three parameters, including VG row numbers, aspect ratios, and channel heights, were optimized. The results revealed that the mixed configuration obtained the best heat transfer at the small channel height but was less efficient in fluid mixing when compared to a larger channel height. The overall performance coefficient improved with increasing channel height and aspect ratio while reducing VG rows. Naik et al. [19] compared the thermal performance of the curved rectangular winglet VG with different arc angles in a rectangular channel. Longitudinal vortices contributed to a higher heat transfer than transverse vortices as the better fluid transport and mixing. The best thermal performance factor was the convex rectangular VG with an arc angle of 30°.

VGs have been extensively investigated in different heat exchangers, such as double-pipe heat exchangers (DPHEs) [20,21,22,23,24,25] and plate heat exchangers [26]. Jawarneh et al. [20] examined the influence of jet VGs with different inlet flow angles of 30°, 45°, and 60° on heat transfer improvement within the inner tube of a DPHEs. The results indicated an 82% increase in the Nusselt number for the inlet flow angle of 60°. Li et al. [21] analyzed the thermal performance of VGs in a double-pipe heat storage system, wherein star-shaped VGs were implemented on the annular side with phase change material (PCM), and the inner side used water as the fluid. The application of VGs inside the PCM region reduced the melting time by 100 min while using VG in the water contributed to a higher heat transfer rate from PCM to water in a shorter time. Aridi et al. [24] studied heat transfer improvement using trapezoidal VGs in a concentric tube heat exchanger, considering VG placement in either the tube or annular side. The study revealed that the highest heat transfer improvement of 97% was achieved when adding VGs to the tube side. Tang et al. [26] investigated the thermo-hydraulic behavior of VGs with various shapes and α in a heat exchanger designed to cool water with oil as the medium. They found that the elliptical winglet VG with an α of 30° in oil and 15° in water achieved the highest thermal performance factor.

Based on the surveyed literature, it is evident that longitudinal VGs have received widespread attention across diverse systems, with a particular emphasis on microchannel applications. However, a noticeable gap persists as limited studies have addressed the application of longitudinal VGs on the annular side of DPHEs. Furthermore, comparative studies between flat and curved VG surfaces are limited and warrant further study. Additionally, the influence of perforations in flat and curved VG structures needs a comprehensive investigation. Considering the gaps in the literature, this study aims to contribute significantly by undertaking an in-depth evaluation of the thermo-hydraulic performance of perforated rectangular and sinusoidal VGs. This investigation will focus on three different α, namely, 15°, 45°, and 75°, within a DPHE system operating under turbulent flow conditions. The selection of angles aims to provide a detailed understanding of the impact of varying configurations on heat transfer enhancement. Moreover, the research will delve into a thorough analysis of the flow characteristics resulting from the implementation of different VGs. This aspect is crucial for unraveling the intricate interactions between fluid dynamics and heat transfer processes in the DPHE system. Moreover, the novelty of this research extends to the exploration of three different spacings (300, 100, and 60 mm) within the optimized VG configuration. The anticipated outcomes of this study could hold the promise of advancing fundamental understanding and offering practical implications for optimizing VG design and elevating the overall performance of DPHE systems.

Methodology

Geometry

Figure 1 shows the physical model of the DPHE with RVGs fixed on the annular side. Here, the cold water passes through the annular side with an inlet temperature of 25 °C, and the hot water flows within the tube in the opposite direction with an inlet temperature of 50 °C. The outer pipe is insulated so that the heat transfer only happens within the system. The total length of the system is 1,700 mm, with a 100 mm section at both ends being thermally insulated, thus making the 1,500 mm section in between the effective test section. The first 100 mm extended section is for the fluids to be fully developed, while the last 100 mm is the area to collect the outlet temperature data and avoid the effect of the thermal boundary layer beyond the 1,500 mm test section. Eleven VGs are fixed in the center part of the annular side with a spacing of 100 mm. The flow rate of the hot side is constant at 10 L/min, while the cold side varies between 10 and 20 L/min, corresponding to Reynolds numbers of 4,759–9,518. Table 1 shows various VG shapes and their dimension details. The width, length, and height for all the VGs are the same. A circular ring with a length of 8 mm and thickness of 0.5 mm is used to support the VG. The sinusoidal shape is drafted with a slight curvature of 1 mm for its amplitude value. With that, this study compares the influence of changing rectangular, sinusoidal faces, hole features, and α on the system’s heat transfer performance and pressure drop.

Fig. 1
figure 1

Dimensions of the studied model

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Table 1 Studied VG shapes and dimension details
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Numerical techniques

The numerical analysis is conducted using a three-dimensional (3D) and steady-state model by employing the ANSYS Fluent software. The conservation equations for steady flows with turbulence are shown in Eqs. (1–3) [27]. To calculate heat transfer and fluid flow behavior, the k–ω turbulence model is utilized and shown in Eqs. (4) and (5) [27]. When using a mesh compliant to y + of about 1, this turbulence can accurately predict the flow behavior for the fluid separation and swirl, which originate from the near-wall regions. To ensure mesh compliance for this turbulence model, the first layer cell thickness is set to ensure the y + value of approximately 1 is achieved for the highest Reynolds number investigated. To handle the complexities of the geometries involved, the coupled algorithm is selected due to its robustness and relatively quick convergence rate. For the discretization of all equations, the second-order upwind scheme is chosen. The simulation is terminated when the residuals of all governing equations reach below 10−6. The velocity-inlet and pressure-outlet boundary conditions are set for the inlet and outlet of each respective fluid channel. The outer pipe wall is fully insulated, and the VG is treated as being thermally conductive. The inlet temperatures for the cold and hot sides are set to 25 °C and 50 °C, respectively. The thermophysical properties of water at different temperatures are extracted from the Engineering Toolbox [28]. Additionally, gravity effects are considered in this study.

Continuity equation:

$$\frac{\partial }{{\partial x_{{\text{i}}} }}\left( {\rho \mu_{{\text{i}}} } \right) = 0$$
(1)

Momentum equation:

$$\frac{\partial }{{\partial x_{{\text{j}}} }}\left( {\rho \mu_{{\text{i}}} \mu_{{\text{j}}} } \right) = - \frac{\partial P}{{\partial x_{{\text{i}}} }} + \frac{\partial }{{\partial x_{{\text{j}}} }}\left[ {\mu \left( {\frac{{\partial \mu_{{\text{i}}} }}{{\partial x_{{\text{j}}} }} + \frac{{\partial \mu_{{\text{j}}} }}{{\partial x_{{\text{i}}} }}} \right)} \right] + \frac{\partial }{{\partial x_{{\text{j}}} }}\left( { - \rho \overline{{u_{{\text{i}}}^{\prime } u_{{\text{j}}}^{\prime } }} } \right) + \rho g_{{\text{j}}}$$
(2)

Energy equation:

$$\frac{\partial }{{\partial x_{{\text{i}}} }}\left( {\rho \mu_{{\text{i}}} T} \right) = \frac{\partial }{{\partial x_{{\text{j}}} }}\left[ {\left( {\frac{\mu }{\Pr } + \frac{{\mu_{{\text{t}}} }}{{\Pr_{{\text{t}}} }}} \right)\frac{\partial T}{{\partial x_{{\text{j}}} }}} \right]$$
(3)

k–ω equation:

$$\frac{\partial }{{\partial x_{{\text{j}}} }}\left( {\rho u_{{\text{i}}} k} \right) = \frac{\partial }{{\partial x_{{\text{j}}} }}\left[ {\left( {\mu + \frac{{\rho k\alpha_{1} }}{{\max \left( {\omega \alpha_{1} ,SF_{2} } \right)\sigma_{{\text{k}}} }}} \right)\frac{\partial k}{{\partial x_{{\text{j}}} }}} \right] + P_{{\text{k}}} - \beta^{*} \rho k\omega$$
(4)
$$\frac{\partial }{{\partial x_{{\text{j}}} }}\left( {\rho u_{{\text{i}}} \omega } \right) = \frac{\partial }{{\partial x_{{\text{j}}} }}\left[ {\left( {\mu + \frac{{\rho k\alpha_{1} }}{{\max \left( {\omega \alpha_{1} ,SF_{2} } \right)\sigma_{\omega } }}} \right)\frac{\partial \omega }{{\partial x_{{\text{j}}} }}} \right] + \frac{\alpha \omega }{k}P_{{\text{k}}} - \beta \rho \omega^{2} + 2\left( {1 - F_{1} } \right)\frac{\rho }{{\omega \sigma_{{{\omega 2}}} }}\frac{\partial k}{{\partial x_{{\text{j}}} }}\frac{\partial \omega }{{\partial x_{{\text{j}}} }}$$
(5)

where \(\sigma_{{\upomega }} = 2, \;\sigma_{{\text{k}}} = 2, \;\beta = 0.075, \;\beta^{*} = 0.09, \;\alpha = 1/9, \;\sigma_{{{\omega 2}}} = 1/0.856.\)

Figure 2 shows the mesh independence study in the model with three SVGs at α of 15° and Reynolds number of 9,518. As Fig. 2a shows, structured meshes are used for the hot fluid and steel pipe, while tetrahedral meshes are applied for the cold fluid. To facilitate the accurate resolution of flow characteristics within the near-wall region, twelve and sixteen inflation layers are added along the inner and outer walls within the cold fluid domain. The mesh size in the VG area is more refined than in other areas. Additionally, the mesh quality is assessed to make sure the skewness is less than 0.88. In Fig. 2b, the Nusselt number increases when the mesh element numbers are raised. The increasing trend becomes smaller when the cell number exceeds 7.5 million. The difference in the Nusselt number is 0.89% when increasing the element numbers from 7.5 million to 11 million. Thus, the 7.5 million mesh is selected to calculate the fluid performance with accurate results and less computational time. The same meshing density is also applied to other cases to maintain similar grid accuracy between simulated cases.

Fig. 2
figure 2

Mesh independence study for the studied model with three SVGs at α = 15° and Re = 9,518

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Parameter definitions

The following equations are employed for evaluating the thermo-hydraulic performance of modified geometries on the annular side of the system.

Equation (6) defines the average heat transfer rate of the hot fluid.

$$Q_{{\text{h}}} = m_{{\text{h}}} C_{{\text{p,h}}} \left( {T_{{\text{h,i}}} - T_{{\text{h,o}}} } \right)$$
(6)

Equation (7) defines the average heat transfer rate of the cold fluid.

$$Q_{{\text{c}}} = m_{{\text{c}}} C_{{\text{p,c}}} \left( {T_{{\text{c,o}}} - T_{{\text{c,i}}} } \right)$$
(7)

Equation (8) defines the average heat transfer rate of the system. Here, the difference in the average heat transfer rate between hot and cold sides is lower than 3%.

$$Q_{{{\text{avg}}}} = \frac{{Q_{{\text{h}}} + Q_{{\text{c}}} }}{2}$$
(8)

Equation (9) defines the average heat transfer coefficient in the cold side.

$$\overline{h}_{{\text{c}}} = \frac{{Q_{{{\text{avg}}}} }}{{\left( {\overline{T}_{{\text{w}}} - T_{{\text{c,b}}} } \right)}}$$
(9)

Equation (10) defines the average Nusselt number.

$$\overline{{{\text{Nu}}}}_{{\text{c}}} = \frac{{\overline{h}_{{\text{c}}} D_{{\text{h}}} }}{k}$$
(10)

Equation (11) defines the Reynolds number.

$${\text{Re}} = \frac{{\rho vD_{{\text{h}}} }}{\mu }$$
(11)

Equation (12) defines the friction factor .

$$f_{{\text{c}}} = \frac{{2\Delta PD_{{\text{h}}} }}{{L\rho v^{2} }}$$
(12)

Equation (13) defines the performance evaluation criterion (PEC) value [29].

$${\text{PEC}} = \frac{{{\text{Nu}}_{{\text{c}}} }}{{{\text{Nu}}_{{\text{c,s}}} }}/\left( {\frac{{f_{c} }}{{f_{{\text{c,s}}} }}} \right)^{1/3}$$
(13)

To validate the accuracy of the simulation results, Petukhov (14) and Sieder–Tate Eq. (15) are utilized to validate the Nusselt number [30, 31]. At the same time, the friction factor is validated by Petukhov and Blasius equations shown in Eqs. (16) and (17), respectively [31, 32]. These correlations are applicable to the turbulent flow in a DPHE.

$${\text{Nu}} = \frac{{\left( \frac{f}{8} \right){\text{Re}} \Pr }}{{1.07 + 12.7\left( \frac{f}{8} \right)^{0.5} \left( {\Pr^{\frac{2}{3}} - 1} \right)}}$$
(14)
$${\text{Nu}} = 0.027{\text{Re}}^{\frac{4}{5}} \Pr^{\frac{1}{3}} \left( {\frac{\mu }{{\mu_{{\text{s}}} }}} \right)^{0.14}$$
(15)
$$f = \left( {0.79\ln \left( {\text{Re}} \right) - 1.64} \right)^{ - 2}$$
(16)
$$f = 0.316 {\text{Re}}^{ - 0.25}$$
(17)

Validations

The simulation model for the smooth case is validated against theoretical correlations, as shown in Fig. 3. The average Nusselt number deviations between the simulation and the theoretical correlations are 9.49% and 9.39% for the Petukhov and Sieder-Tate equations, respectively. Meanwhile, the average friction factor differences between the simulation and Petukhov and Blasius equation are 2.86% and 4.15%, respectively. The difference between the numerical simulation and the correlation equations can be attributed to grid spacing and the ideal conditions imposed using ideal boundary conditions in numerical simulation, which can provide perfect heat flux and adiabatic regions. Since the error from the numerical simulation is found to be lower than 10%, the current numerical model can obtain reasonably accurate results to evaluate the heat transfer performance of DPHEs.

Fig. 3
figure 3

Numerical validations with theoretical correlations

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Results and discussion

The presence of the VG generates secondary flow and vortices, which can be visualized from the numerical analysis. Distinct vortex structures arise from different VG configurations, crucially affecting thermo-hydraulic performance. Thus, studying flow patterns is essential for understanding the heat transfer mechanism and pressure drop characteristics.

Flow characteristics of RVGs

Figure 4 shows the velocity contours and streamlines at the normal cross section located 30 mm downstream of the RVG and the cross section parallel to the flow direction on the left and right, respectively. In cases without holes, as shown in (a), (b), and (c), a recirculation zone is observed downstream of the RVG, where the fluid tends to circulate in a small region without moving forward. The intensity of the transverse vortices increases as α increases from 15° to 75°. Moreover, the flow starts to twist after the recirculation zone. Due to the flow mixing among the leading and trailing edges, as well as the RVG tips, a large and substantial longitudinal vortex is formed. When four RVGs are placed in one row, four large longitudinal vortices form in the flow domain. The two close vortices interact to cause a low-pressure drop zone in between, where the velocity is the lowest. When adding holes to these RVGs, as displayed in (d), (e), and (f), reduced flow velocities are found for all the cases. The transverse vortex is divided into small vortices when passing through the holes. The longitudinal vortices caused by the leading edge are weaker than the no-hole cases, resulting in the flow separation downstream of the RVG for the 15° and 45° cases. Two small secondary vortices are observed in between the large vortices with a counter-rotated direction. However, no separated flow is observed in the 75° case. It is due to the increased slope with a high blockage generating stronger longitudinal vortices at the leading edge compared to the 15° and 45° cases.

Fig. 4
figure 4

Velocity streamlines and contours of various RVG cases at different cross sections with Re = 9,518

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Figure 5 presents the temperature contours on the outer wall of the steel pipe for different RVG cases. Variations in local wall temperature are observed among the cases, which can be attributed to the different vortex patterns, as depicted in Fig. 4. The longitudinal vortices are sufficiently robust to reduce the global wall temperature. However, the influence of the benefit diminishes as the flow passes further downstream. The region between the VGs shows a higher wall temperature (indicated by the orange color) compared to other areas that correspond to the low-velocity zone. In essence, the strength of the longitudinal vortices decreases as the flow moves further downstream of the VG. For cases without holes, as shown in (a), (b), and (c), the 15° case exhibits a higher wall temperature compared to other cases, where the wall temperature is cooled down over a shorter distance and small surface area, facilitated by the diminished strength of the weak longitudinal vortices. Conversely, the 45° case, characterized by more robust longitudinal vortices, effectively reduces the wall temperature over an extended distance. The 75° case cools the wall across a considerable surface area near the VG more effectively. However, its effectiveness is rapidly reduced due to the low twisting flow at this angle. When introducing holes to these cases, illustrated in (d), (e), and (f), the wall temperature near the RVG for these hole cases is higher than the no-hole cases due to the reduced flow velocities. The longitudinal vortices caused by the RVG are also less effective in cooling the global wall temperatures. As a result, the overall heat transfer is reduced when holes are added to the RVGs. Additionally, the 75° case with holes displays a higher global wall temperature compared to the 45° case with holes. This discrepancy can be attributed to the hole direction being nearly parallel to the flow, leading to a more pronounced reduction in transverse vortices strength and, consequently, weaker longitudinal vortices.

Fig. 5
figure 5

Steel pipe wall temperature contours in various RVG cases at Re = 9,518

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To elucidate the impact of vortex structures on wall temperature variations within the VG area, the three-dimensional velocity streamlines, and temperature contours for different RVG cases are depicted in Fig. 6. Turbulence is consistently observed when the flow traverses the VGs in all cases. Meanwhile, a twisting motion emerges after the VG, leading to the generation of the longitudinal vortices within the flow channel. The cases without holes shown in (a), (b), and (c) exhibit secondary flow characterized by the development of the localized transverse and longitudinal vortices in the VG regions. The 15° case shows the smallest transverse vortices due to the minimal obstruction of the VG in the flow direction. However, as α increases, larger transverse vortices appear, with the 75° case exhibiting the greatest intensity for the transverse vortices among the tested cases. Simultaneously, the flow undergoes twisting motion after the trailing edges, leading to localized enhanced heat transfer in the 15° case. Conversely, for the 45° and 75° cases, the transverse vortices do not directly improve the heat transfer as the heat is trapped locally. The flow velocities are increased between the transverse vortices, where the wall is cooled down more efficiently due to better mixing with the freestream flow. Notably, the introduction of holes diminishes the transverse vortices for all cases, as presented in (d), (e), and (f). In the 15° case, the presence of holes introduces longitudinal vortices, contributing to a reduction in wall temperature near the VG. The 45° case with holes shows small transverse vortices, providing minimal improvement in heat transfer. However, the enhanced flow velocity between transverse vortices and local twisting motion contributes to enhanced local heat transfer. A higher local wall temperature between the VGs is observed for the 75° case, as the flow velocity is not significantly increased in this area when the hole direction is nearly parallel to the flow.

Fig. 6
figure 6

Three-dimensional velocity streamlines and temperature contours of various RVG cases with Re = 9,518

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Thermo-hydraulic performance of RVGs

Figure 7 shows the average wall temperature variations across various cases within the RVG regions at the Reynolds number of 9,518. The overall wall temperature exhibits a gradual increase in the flow direction, attributed to a diminishing difference between the cold and hot fluid. All RVG cases consistently demonstrate lower wall temperatures compared to smooth pipe due to the presence of the VG. In Fig. 7, two vertical lines delineate the VG locations from x = 0.746 m to x = 0.754 m. When the flow passes through the VG at x = 0.746 m, a marked reduction in the wall temperature is observed, as the conductive VG introduces additional surface area, facilitating accelerated interaction between the cold and hot fluid. Figure 6 corroborates this observation, depicting lower wall temperature contours within the VG area for all cases relative to other regions. The 75° case exhibits the lowest wall temperature within the VG, which can be attributed to larger faces in the leading-edge area that interact with the flow. Additionally, the perforations provide more impact on the wall temperature for the high α. The wall temperature increases in the trailing-edge area due to the sudden reduction in fluid velocity. A sharp reduction in wall temperature occurs at x = 0.754 m, where the 45° case without holes presents the lowest temperature, driven by the pronounced twisting motion at this angle. However, the 75° case without holes does not show a decreasing temperature because the large transverse vortices trap more heat in this area. In contrast, the 15° case with holes obtains the highest wall temperature, reflecting a weak twisting motion. Subsequently, the wall temperature gradually increases beyond the 0.76 m region despite the influence of global longitudinal vortices. The 45° and 75° cases present lower temperatures owing to robust longitudinal vortices. Among the cases with holes, the 45° case demonstrates more efficient wall cooling, presenting the lowest wall temperature compared to other hole cases. In contrast, the 75° case with holes achieves more efficient wall temperature reduction near the VG compared to the 15° case with holes, while the longitudinal vortices for this case exhibit reduced strength comparatively along the channel.

Fig. 7
figure 7

Wall temperature variations of various RVG cases at different locations and Re = 9,518

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Figure 8a–c shows the comparison of Nu/Nus, friction factor, and PEC in various RVG cases, respectively. As shown in (a), the 75° case without holes achieves the highest Nusselt number among all the cases, exhibiting a remarkable 42.4% improvement in heat transfer than the plain pipe at the Reynolds number of 5,711. However, this enhancement reduces to 23.66% when adding holes in RVGs at the same α and Reynolds number. It is because the hole almost aligned to the flow direction diminishes the transverse and longitudinal vortices. Similarly, the 15° and 45° cases without holes also display higher heat transfer enhancement compared to the case with holes. The decrement in Nusselt number for these two cases is comparatively less than that observed in the 75° case due to the slanted orientation of the holes to the flow direction. On the other hand, the 45° case demonstrates superior performance among the cases featuring holes. It can be elucidated by the presence of more robust longitudinal vortices, which contribute to a lower wall temperature, as illustrated in Fig. 7. The lowest Nusselt number is found in the 15° case with holes at the Reynolds number of 5,711, with 20.8% enhancement over the plain pipe.

Fig. 8
figure 8

Comparison of thermo-hydraulic performance of various RVG cases for different Reynolds numbers

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Observing the friction factor trends in Fig. 8b, the 75° case without holes exhibits the maximum friction factor, approximately two times higher than the plain pipe at the Reynolds number of 5,711. However, the friction factor significantly drops when adding holes in this case. At the same time, the 45° case also displays the reduced faction factor by adding holes. A similar trend is observed for the 15° case with and without holes as the hole is nearly perpendicular to the flow direction, where the friction factor is increased by around 30% compared to the plain pipe at the Reynolds number of 5,711.

As for the PEC variations in Fig. 8c, all the cases display a reduction in PEC with increasing Reynolds numbers. The 15° case without holes presents the highest PEC with the maximum value of 1.16 among the cases at the Reynolds number of 5,711. Although the 75° case without holes shows the highest Nusselt number, the higher friction loss leading to the PEC is lower than 15° and 45° without hole cases. When adding holes in the RVG, the PEC reduces significantly for the 15° and 75° cases, as the reduced friction factor cannot compensate for the decrease in heat transfer. Conversely, the 45° case with holes displays a slightly higher PEC than the no-hole case.

Flow characteristics of SVGs

Figure 9 shows the velocity contours and streamlines at the normal cross section located 30 mm downstream of the SVG and the cross section parallel to the flow direction on the left and right, respectively. Similar to RVGs, transverse vortices originating from SVGs convert into longitudinal vortices. Notably, the SVG induces more vortices compared to the flat RVG shown in Fig. 4. The increased number of transverse vortices results in intricate longitudinal vortex structures within the flow domain as these transverse vortices interact with each other. In the 15° case (a), four large longitudinal vortices are observed because of the fluid mixing and twisting among the leading and trailing edges, as well as the SVG tips. At the 45° case in (b), two pairs of vortices are generated by the two adjacent SVG. One SVG produces the longitudinal vortex center near the wall, while the neighboring SVG generates the longitudinal vortex with the center away from the wall. The one close to the wall is more powerful, leading to the surrounding flow moving in the same direction. It goes to the near wall area and connects to generate a large vortex with two different vortex centers. In the 75° case (c), the leading edge causes the longitudinal vortex center to be far from the wall, while the trailing edge results in the vortex center being close to the wall. Four pairs of vortices are observed in the normal cross section with counter-rotated directions. When holes are introduced for all cases, the transverse vortices formed downstream of the SVG diminish, thus reducing the strength of the longitudinal vortices, with the 15° case being an exception. The flow velocity remains constant for a longer distance due to the reduced blockage effect. Additionally, two small vortices with counter-rotating directions are observed between the large vortices caused by the secondary flow in the low-pressure zone. In the 45° case with holes, four pairs of longitudinal vortices are observed in the transverse cross section with local fluid mixing. Two adjacent SVGs cause two large longitudinal vortices to interact with each other with counter-rotated directions. A low-pressure drop zone is generated between the large longitudinal vortices, causing two small secondary vortices. A similar phenomenon can be observed for the 75° case with holes, where two neighboring SVGs with smaller spacing create two longitudinal vortices with different strengths, resulting in secondary vortices with different intensities.

Fig. 9
figure 9

Velocity streamlines and contours of various SVG cases at different cross-sections with Re = 9,518

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Figure 10 shows the outer wall temperature contours of a steel pipe in various SVG cases at the Reynolds number of 9,518. Like RVGs, the vortices caused by the SVG exhibit robustness in cooling the region near the SVG, and the low-pressure zone occurs between the SVG placed closely. The high-temperature spots with yellow color appear near the SVG as the tiny transverse vortices trap the heat, leading to localized hotspots. Furthermore, the longitudinal vortices formed in the 45° case are more powerful in cooling the global wall temperature compared to the 15° and 75° cases. When holes are added in the SVGs, the 45° and 75° cases show a higher average temperature compared to the no-hole cases due to the weakened strength of the longitudinal vortices caused by the holes. In contrast, the 15° case with holes can cool down the wall temperature over a longer distance compared to the no-hole case. This improvement is attributed to the increased flow velocity shown in Fig. 9, which intensifies the strength of the longitudinal vortices.

Fig. 10
figure 10

Steel pipe wall temperature contours in various SVG cases at Re = 9,518

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To elucidate the thermal characteristics in the vicinity of the VG, Fig. 11 provides the three-dimensional velocity streamlines, and the wall temperature contours in SVG cases at the Reynolds number of 9,518. The curved structure of the SVG induces a pressure differential behind the VG zones, leading to the formation of multiple transverse vortices. In particular, the 15° case presented in (a) exhibits smaller transverse vortices at the trailing edges, with an augmented α resulting in larger transverse vortices. The 75° case illustrated in (c) displays the largest transverse vortex zone among the cases. Like the RVG cases, these transverse vortices indirectly enhance the local heat transfer by accelerating flow speeds between the vortices. The introduction of perforations for all cases results in a notable reduction in the sizes and strength of the transverse vortices. Consequently, the augmentation of flow velocities decreases, and the local heat transfer in cases with holes is lower compared to those without holes. A twisting flow pattern is also observed behind these transverse vortices for all cases. The 15° cases (with and without holes) and the 45° case without holes demonstrate superior flow twisting along the channel compared to other cases. In contrast, the 75° case with holes does not improve local heat transfer efficiently due to the diminished transverse vortices.

Fig. 11
figure 11

Three-dimensional velocity streamlines and temperature contours of various SVG cases with Re = 9,518

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Thermo-hydraulic performance of SVGs

Figure 12 illustrates the wall temperature variations across different SVG cases within the VG area when considering the location x ranging between 0.74 m and 0.84 m for the Reynolds number of 9,518. The two vertical lines represent the VG location between 0.746 m and 0.754 m. As with the RVG case, the smooth pipe exhibits higher wall temperatures compared to all investigated SVG cases. A sudden reduction in temperature occurs as the flow encounters the leading edges of the SVG, followed by an increase in temperature at the trailing edge, where flow velocity reaches a minimum. Among the tested cases, the 75° case without holes yields the lowest wall temperature here due to the larger VG surface at the leading edge that enhances mixing in the flow. Conversely, the introduction of holes reduces the VG surface area, leading to increased wall temperatures. Notably, perforations have negligible effects on the wall temperature in the 15° case due to the consistent surface area of the leading edge. Beyond the VG when x is more than 0.754 m, transverse vortices induce enhanced flow velocities between them, resulting in increased local heat transfer. The 45° case without holes presents the lowest local wall temperature, which can be attributed to robust transverse vortices and a twisting flow, whereas the 75° case with holes shows the highest local wall temperature due to the weakened transverse vortices, as shown in Fig. 11. Subsequently, wall temperatures gradually increase in the channel for all tested cases. The 45° case without holes shows the strongest longitudinal vortices and lowest wall temperature among all tested cases. Among cases with holes, the 15° case proves more effective in cooling the wall compared to the other tested cases.

Fig. 12
figure 12

Wall temperature variations of various SVG cases at different locations and Re = 9,518

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Figure 13 shows the Nu/Nus, friction factor, and PEC variations for different SVG cases. As shown in Fig. 13a, the 45° case without holes displays the highest average Nu/Nus, with an improvement of 34.59% at the Reynolds number of 5,711. The enhancement is significantly reduced to 28.31% when holes are added to the SVG. The 15° and 75° cases display similar Nu/Nus at the same Reynolds number. However, the 15° case with holes presents a higher heat transfer compared to the no-hole case, while the 75° case with holes dramatically reduces Nusselt number enhancement. These trends can be explained by the vortex patterns and wall temperatures depicted in Figs. 10 and 11.

Fig. 13
figure 13

Comparison of the thermo-hydraulic performance of various SVG cases in different Reynolds numbers

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For the friction factor variations in Fig. 13b, similar trends are observed between the SVG and RVG cases. All the cases show a higher friction factor when compared to the smooth pipe in the studied Reynolds number ranges. The friction factor increases with the increase in α and reduces as the Reynolds number increases. On the one hand, the 75° case shows the highest friction factor, around twice as high as the smooth pipe at the Reynolds number of 5,711. However, a significant reduction in friction factor is observed for the 75° case when holes are added. On the other hand, adding holes in the 45° case marginally reduces the friction factor. A similar friction factor can be observed in the 15° case when comparing both with and without holes. Therefore, the hole is critical in reducing the pressure drop for high α cases.

Figure 13c shows that PEC decreases as the α increases, with a similar outcome occurring with the increasing Reynolds numbers. The 15° case with holes shows the maximum PEC of 1.18 at the Reynolds number of 4,957 due to the increased flow velocity causing stronger vortices compared to the no-hole case. A reduced Nu/Nus is observed for other cases when holes are added. Although the pressure drop is also reduced, the heat transfer loss is still greater. Therefore, adding holes reduces the PEC for the 45° and 75° cases. The 75° case with holes yields the lowest PEC of 0.95 at the Reynold number of 9,518.

Spacing effect in thermo-hydraulic performance

Through a comparative analysis of the thermo-hydraulic performance of various cases, the SVGs-hole case achieves the highest PEC value with a spacing of 100 mm. This section presents the Nusselt number, friction factor enhancement, and PEC variations for three SVG-hole cases with 60, 100, and 300 mm spacings, as shown in Fig. 14. In Fig. 14a, the Nu/Nus significantly increases when reducing the spacing from 300 to 60 mm, reaching a maximum value of 1.46 at the Reynolds number of 4,795. At a high spacing of 300 mm, Nu/Nus remains relatively constant within the studied Reynolds number ranges. However, as the spacing decreases to either 100 or 60 mm, the Nu/Nus drops significantly with the increasing Reynolds numbers. This behavior can be attributed to the sufficient space for vortices to evolve and cool down the wall in high-spacing cases, while the vortices tend to break down within shorter distances in low-spacing cases. The efficiency of vortices is also reduced as Reynolds number increases. Figure 14b demonstrates a substantial increase in friction factor as the spacing decreases from 300 to 60 mm, resulting in friction factor enhancements ranging from 9.42% to 73.91%. In Fig. 14c, the PEC value exhibits notable improvement as the spacing decreases from 300 to 100 mm. The highest PEC value of 1.22 is achieved with a spacing of 60 mm and a Reynolds number of 4,795. Additionally, the PEC decreases as the Reynolds number increases, with the reduction being more pronounced for low-spacing cases compared to high-spacing cases.

Fig. 14
figure 14

PEC variations at different spacings and Reynolds numbers for the SVGs-hole case

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Conclusions

This study investigated the thermo-hydraulic performance of rectangular and sinusoidal VGs, both with and without holes installed on the annular side of a DPHE. The VGs are examined under varying α (15°, 45°, and 75°), spacing (60, 100, and 300 mm), and Reynolds numbers. The SVG case showed more vortices compared to the RVG case due to its curved design. Comparing the heat transfer enhancement for both VG types at the same α, the RVG case demonstrated a better heat transfer compared to the SVG case, except for the case with α of 15°. The configuration with a low α and slight curvature increased flow velocities and vortex strength, contributing to higher heat transfer. However, adding holes reduces heat transfer for both VG types, except for the SVG case at the α of 15°, where the flow velocities were enhanced when passing through the holes. Furthermore, the presence of holes had a significant impact on the pressure drop for both VG types for the α of 45° and 75°, while no substantial difference was observed for the α of 15°. This study revealed that the 75° RVG case without holes achieved the highest Nusselt number enhancement of 42.4% at the Reynolds number of 5,711, whereas the highest PEC was 1.18 for the 15° SVGs-hole case. Additionally, the spacing had a considerable effect on heat transfer behavior. When a high spacing of 300 mm was used, the Nusselt number enhancement remained constant despite the increasing Reynolds numbers, while a significant drop was observed for the low-spacing case. The highest PEC value of 1.22 was achieved for the 15° SVGs-hole case with a spacing of 60 mm and a Reynolds number of 4,759.