Effect of modifications of auxiliary surface attached to a rectangular vortex generator

Abstract

Keywords: Thermal performance factor; vortex generator; conjugate heat transfer; primary vortex; auxiliary surface

1. Introduction

In the modern world, every technological development demand sustainability. Reduction of system size and optimization of its energy consumption are two major factors toward such goals. However, in most industries, it is observed that power consumption increases along with the demand for smaller and lighter systems. Therefore, the miniaturization of systems with improved efficiency is the need of the hour. Higher energy consumption demands an effective heat dissipation system to avoid any thermal failure. Fins and vortex generators (VGs) are two popular components of such heat dissipation systems. Implementation of fins augments surface area and subsequently improves heat transfer. The use of VGs gener-

ates various helical fluid motions to enhance the convective thermal performance. This method is particularly effective and thus requires serious consideration. The effectiveness of using VGs to improve heat transfer is reliant on various factors, including geometrical shapes [1-4]. Samadifar and Toghraie [5] studied the impact of different VGs, viz., RVG, wavy VG, etc. The investigation has been carried out numerically using the finite volume method (FVM), on a finplate heat exchanger with a triangular channel cross-section. The results show that the RVG improves the thermal performance of the system by 7%. However, the presence of VG in o the system results in a higher-pressure drop. A 45 angle of attack (φ) is found to provide the best performance, with height

Submitted: 15 March 2024 ; Accepted: 12 June 2024 This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkılıç Published by Yıldız Technical University, İstanbul, Türkiye This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

of the VG contributing to the thermal enhancement. Qian et al. [6] studied the heat exchangers with rectangle winglet VGs. The study found that the rectangle winglet helped in strengthening the weak areas of flow at the rear side of the tubes. This improves the thermal performance and resistance characteristics of the system. Biswas and Chattopadhyay [7] numerically studied the delta wing VGs in a rectangular channel. The Navier–Stokes equations and the energy equation were used to analyze thermal exchange between the fluid and the channel wall. The impacts of pierced holes, and Reynolds number (Re) were investigated. The average Nusselt number (Nu) is found to increase by a maximum of 34%. It was further reported that o the improvement in thermal performance occurs at φ = 26 . Numerous works reported the application of VGs in heat exchangers with improved thermal performance. Different shapes of VGs have been tested using both numerical and experimental studies. The results indicate improved thermal behavior and better thermo-hydraulic performance. In addition, the placement and the orientation of the VGs are found to affect performance further. Luo et al. [8] focused on a new combination of a wavy fin and VG for heat exchangers to increase thermo-hydraulic performance. The findings indicate that the combination of VGs with wavy fins resulted in an increase in Nu by up to 33%. The thermal performance factor is also improved by a maximum of 26.4%. The study investigated different flow regimes and the impact of corrugation angles along o with φ. In the laminar flow regime, the configuration with φ = 45 provides the best thermal performance. Sarangi and Mishra [9], in a separate study, evaluated the thermal behavior of rectangular winglet pairs (RWP). Positioning RWPs in a common-flow-up configuration enhances the heat transfer rate near the central tube. o Thermal performance improves with φ up to 20 , which resulted in a 60% increase in Nu. The winglets are found to perform better, when placed at 3.201 mm upstream in the streamwise direction and 7.2 mm spanwise. However, increased winglet numbers and upstream placement result in higher pressure drops. Thermal performance increased by 12%, 37% and 47%, with one, two and three winglet pairs, respectively. Modi and Rathod [10] studied different rectangular winglet VGs (RWVGs) in heat exchangers, with different Re, varying between 400 and 1000. Compared to a reference case without VGs, RWVGs (wavy-up, wavy-down, curved-up, curved-up, curved-down) show an enhancement of heat transfer. Results show that the wavy-up setup provided the highest Nu. This indicates an increased thermal performance. However, curved-down RWVGs show a better balance in terms of thermo-hydraulic performance. In a separate study, Fiebig et al. [11] investigated wing-type VGs in heat exchangers for Re = 600-2700. Delta winglets were used in both inline and staggered tube systems. In the case of inline systems, VGs enhance thermal performance by up to 65%. However, these devices also increase the friction factor (f) by up to 45%. In the staggered setup, the thermal performance is found to increase by 9%. Gholami et al. [12] studied wavy rectangular winglets for a potential increment in thermal exo change. The Re varied between 400 and 800, with the φ as 30 . The

results show that the wavy rectangular winglet improves the thermal performance over the traditional VGs. The wavy-up setup improves thermal exchange better, with additional flow losses. Pressure drop is noticed with the wavy-down setup. In a different study, Sharma et al. [13] investigated a single triangular shaped winglet VG in a heat exchanger. The authors used triangular structures as auxiliary fins. The VGs were placed on the top and bottom parts of the system under a laminar flow condition. The results show that adding o thickness, along with tilting of the VG at 45 , enhances heat transfer by 19.7%. However, it also increases the pressure loss by 7.8%. This VG configuration reduces the size of the heat exchanger by almost 50%. Gonul and Okbaz [14] conducted a study on how VGs impact thermal performance. Different VG configurations were tested to enhance the flow structure and heat transfer efficiency. Higher vortex intensity and longer VGs are found to affect heat transfer. The expansion of secondary flow regions in the microchannel with an increasing number of VG pairs was also studied [14]. A maximum heat transfer enhancement of 230% was observed, with an increment in pressure loss of 950%. The maximum thermo-hydraulic performance factor was obtained at around 1.38. Välikangas et al. [15] studied delta winglet VGs for a possible enhancement in thermal characteristics. The study reported that the presence of VGs helps in forming longitudinal vortices and improves thermal exchange. It also shows reduced flow losses. The improved design yields a 5.23% improvement in overall performance as compared to the plain fin. Researchers today emphasize the use of longitudinal VGs (LVGs) more than ever because of the potential of these devices to increase heat transfer efficiency in thermal systems. The LVGs generate vortices that disrupt the thermal boundary layer and enhance fluid mixing. Therefore, LVGs are considered as one of the effective solutions for improving thermal performance [16-18]. Liu et al. [19] investigated LVGs inside a rectangular microchannel to study the thermal behavior at Re up to 1200. The study reports a heat transfer enhancement of 9-21% for the laminar regime and of 39-90% for the turbulent regime. It was further reported that pressure losses are also found in the ranges of 34-83% for the laminar regime and 61-169% for the turbulent regime. Wang et al. [20] investigated optimizing an LVG in a circular tube under laminar flow. By varying spacing length, central angle, and slice height, significant enhancements in the Nu and f were found. Ebrahimi et al. [21] conducted numerical investigations on rectangular microchannels with LVGs. The analysis considers single-phase laminar flow. Conjugate heat transfer with temperature-dependent thermo-physical properties is taken into consideration. The study shows a 2-25% increase in the mean Nu and a 4-30% increase in the f with LVGs compared to smooth microchannels, over Re = 100-1100. Various VG geometries, including curved delta winglets, curved rectangular winglets and others, have been analyzed for heat transfer performance in various literature. Zhou and Feng [22] experimentally compared plane and curved winglet VGs of rectangular, trapezoidal, and delta geometries. The study was carried out both with and without punched holes. The curved winglet VGs show

better thermal performance, with lower pressure loss than that of the plane winglet VGs. This was noticed in both laminar and turbulent flow. The curved delta winglet pairs yield a better result. On the other hand, the VGs with punched holes improve thermo-hydraulic performance and reduce the flow resistance. It has been observed that the holes placed at lower and central positions of the VG improve the heat transfer rate and the overall thermo-hydraulic performance. Al-Dulaimi et al. [23] investigated the performance of detached square VGs inside a square duct at Re = 5000. The study varied several parameters such as the blocking ratio, φ, VG number and aspect ratio. The work was carried out numerically using ANSYS Fluent 15. The results indicate that RVGs increase heat transfer by up to 40% due to increased turbulence. A better heat transmission is achieved with the blocking ratio. Thermal performance increases by 17% and 28% for one and three VGs at a blocking ratio of 0.2. o At φ =45 , VGs provide an improved heat transfer. However, higher aspect ratios reduce the heat transfer rate. Mundhe and Bindu [24] investigated the heat response and flow resistance of Conical Offset VGs (COVGs). The study was carried out in a heated steel pipe with turbulent airflow. Numerical simulations were conducted for Re ranging from 4000 to 50000. Both pitch to o o o diameter ratios and φ (15 , 30 , 60 ) were varied. A COVG at φ = o 60 shows the maximum thermal enhancement. The study reports that the experiment was conducted with different pitches relative to diameter values (1.18, 1.97, 3.94). Nu increased from 3.46 to 6.7 because of increased mixing of the fluid near the wall surface. Zhou and Ye [25] conducted an experimental study on a curved trapezoidal winglet (CTW). It was compared with traditional types of VGs, i.e., rectangular, trapezoidal, and delta winglets. The study reported that the delta winglet performs better both in laminar and transitional flows. However, the CTW shows an improvement in thermo-hydraulic performance in the case of fully turbulent flows. This is due to its streamlined shape and low pressure drop. Parametric study o o on CTW indicates that smaller φ at 0 and 15 enhances its performance. A greater curvature of 0.5 and a higher inclination angle of o 20 enhance its performance. The study also reported that double rows of CTW do not improve thermo-hydraulic performance due to increased pressure drop. Therefore, optimized spacing between the rows is necessary. Gentry and Jacobi [26] performed an analysis of delta wing VGs for flow over a flat plate at low Re. The study showed an increment from 50% to 60% in heat and mass transfer. The optimal delta-wing geometries were identified for various values of Re. Wing chord size was used in the analysis. The results show that stronger vortices formed at the boundary layer border. These vortices pulled the outer flow into the boundary layer and reduced its thickness. Xu et al. [27] investigated thermal behavior in a circular tube with the help of VGs. The flow behavior was also analyzed inside the tube. The study considered various parameters such as φ, blockage levels and a few other related geometric variables. The experiments covered airflows with Re = 6000-33000 under uniform heat input. The findings indicate that heat transfer decreases with pitch ratio. However, it increases with Re, φ and blockage levels. The study showed that heat transfer enhancement is about twice that of

the base case. The maximum improvement in thermal performance is 1.45. Salhi et al. [28] analyzed natural convection in a nanofluid (water with silver or titanium dioxide) in an inclined wavy cavity using FVM. A parametric analysis of variables such as volume fraction, Rayleigh number (Ra), etc., was performed. It was found that the presence of nanoparticles improves heat transfer. The study further reported improved results at higher volume fractions and Ra. The modification of the wavy surface geometry also helps in optimization of thermal performance. In recent years, researchers have been finding ways to improve heat transfer performance using nanofluids. Many studies apply nanofluids to natural convection systems. These investigations are typically conducted in laboratory flasks to study thermal performance. Chafai et al. [29] studied buoyancy-driven flow of nanofluids in a tilted flat-bottom flask using the FVM. Copper (Cu) and aluminium oxide (Al₂O₃) nanoparticles with various volume fractions were considered. The results show that Cu nanofluids provide better heat transfer than Al2O3 nanofluids, especially at higher nanoparticle volume fractions and Ra. Baiti et al. [30] investigated turbulent mixed convection of a graphene-carbon nanotube hybrid nanofluid inside a round-bottom flask. Thermal performance was found to be impacted by Ra and the nanoparticle volume fraction. However, the agitator speed showed a minor impact. Salhi and Chafai [31] numerically investigated laminar natural convection in a square cavity filled with a water-based hybrid nanofluid containing Ag and Ag-TiO2 nanoparticles. The cavity featured one heat source and one heat sink with adiabatic walls. The FVM was used to study parameters such as volume fraction of nanoparticles. The type of hybrid nanofluid was also studied. It was found that thermal performance improves when the nanoparticle volume fraction is raised. Previous studies have investigated the influence of VGs on heat transfer in solar air heaters. These studies show that VGs improve mixing, which in turn increases heat transfer and overall thermal efficiency [32-35]. Alnakeeb et al. [36] investigated a corrugated plate solar air heater fitted with VGs. A 3D numerical model was used to analyze various shapes of VGs, such as rectangular, trapezoidal, and delta-shaped VGs. The study was conducted at various values of φ. The results indicate better thermo-hydraulic performance than that of a corrugated plate alone. Better performance was obtained o with RVGs at Re = 7000 and φ = 30 . The study showed that VGs with corrugated plates improve solar air heater efficiency. Hu et al. [37] studied the optimization of solar air heaters by integrating oneeighth sphere VGs into the absorber plate. An electric plate was used to mimic the solar heat input. The study was conducted to find the impact of deflection angle and pitch. Improved performance was o achieved at a deflection angle of 180 . A performance factor of 2.03 was also obtained in the study. In comparison to the plain duct, Nu and f are found to increase by factors of 2.45 and 1.96, respectively. The configuration shows an effective method of enhancement for

solar air heaters. Sawhney et al. [38] conducted experiments on a solar heater to study the effect of VGs on thermal performance. The study focused on wavy up delta winglet VGs. It was found that placing the VGs at a suitable location in the fluid domain improves system performance. The study further reported that the thermal o performance is best at φ = 60 . This results in an increase in Nu by approximately 223% compared to a flat plate, with a particular setup at Re = 4000. Tian et al. [39] investigated novel winglet VGs within heat exchangers used for flue gas. It was found that VGs increase surface heat flux by 15.8% and pressure drop by 78.9%, compared to plain fins. Rectangular winglets perform better, increasing surface heat flux by a significant amount. Fahad et al. [40] numerically studied five novel VG shapes in a rectangular channel to improve thermal behavior. The angular orientations of the VGs were also considered in the study. Recently, studies on VGs with perforations have gained a lot of focus and have been shown to provide enhanced thermal performance. This technique has improved system efficiency by enhancing fluid mixing and flow dynamics. The effects are visible in the system in terms of thermo-hydraulic efficiency and pressure drop characteristics [41-44]. Saini et al. [45] investigated curved delta winglet VGs (CDWVGs), in both perforated and solid forms. The work was done to enhance thermal performance in heat exchangers. Thermo-hydraulic performance across Re = 400-2000 was evaluated with the help of 3D numerical analysis. Among the cases considered, CDWVGs with 6 circular perforations are found to provide the best efficiency. Studies show that modifying the surface of VGs may enhance heat transfer. Kashyap et al. [46] conducted a study on convex and concave surface modifications on both sides of VGs, which resulted in the primary vortex being placed in the right position on the downstream side of the flow, making it stronger than RVG. The cross-sectional area at the vertical belly part was found to be crucial in enhancing heat transfer, while the width towards the upper and lower ends did not affect much. Several concave profiles on the leading side of the VG enhance the heat transfer, whereas convex profiles on the leading and trailing surfaces do not help to improve the thermal performance. Another study introduced modifications to the trailing surface in the form of a step, which increases the intensity of the primary vortex, resulting in increased heat transfer [47]. The VG with a concave semi-circular texture showed the greatest increase in the Nu at 14.4%, while the coefficient of pressure increases by 3.24%. Apart from surface modifications, different modifications, such as an attached surface on the RVG, have enhanced heat transfer significantly. Kashyap et al. [48] numerically investigated on an extended surface attached to the VG. It reported an increase in Nu of 8.9%. The authors [48] indicated that the presence of an additional surface helped to produce stronger vortices with increased shear within the flow, resulting in an improved heat transfer rate. The literature demonstrates that a suitable modification of VG enhances the heat transfer rate by strengthening vortices. At the same time, extended surfaces are known to improve heat transfer rates. However, detailed parametric studies on VGs with auxiliary surfaces

are still limited. Very few studies have analyzed how these auxiliary surfaces affect heat transfer under different configurations. Therefore, this work focuses on the influence of newly designed trapezoidal-shaped auxiliary surfaces on heat transfer. In the current study, the numerical model is initially validated using existing literature data for an RVG fitted on a plate. The primary objective is to explore the effect of four different geometric parameters on thermal and flow behavior of the system. The volumes of the VGs across all the studies are kept constant to facilitate a justifiable comparison. This study focuses on improving heat transfer for a wide range of engineering applications. The new design may help in enhancing the performance of heat exchangers, HVAC systems, car radiators, aerospace cooling systems, power generators, and chemical reactors. It will also offer better cooling for high heat flux components in electronics by disrupting the boundary layer and mixing fluids more effectively.

2. Geometry and formulations

The rate of heat transfer is greatly influenced by the addition of an extended surface. The type and the orientation of the extended surface play an essential role in quantifying the heat rejection rate. In the current study, a VG is placed over a heated surface. Figure 1a shows the schematic diagram of an array of rectangular VGs in a channel with flow between two parallel plates. Owing to the symmetry of the system, the computational domain (Figure. 1b) is selected in a way to accommodate only a single VG over the base plate. To analysis, a computational domain of Lx × Ly × Lz is taken into consideration (Figure. 1b). A rectangular vortex generator (hp × lp × tp) (Figure. 1c) is placed at an angle of φ to the incoming flow over a heated flat plate (Lx × Lz) (Figure. 1b). With an aim to improve the thermal performance of the system, the principal part (PP) of an RVG is modified with the help of an extended auxiliary part (AP) of trapezoidal shape. Figure 1d shows various configuration parameters of such a modified RVG (MRVG). The trapezoidal AP is placed over the PP at a height of ha. With an angle of (α1 = α2), the base of the trapezoid is shared with PP, and is same as lp for all the cases. The other base, the width and the thickness of the trapezoid are taken as la, w, ta. The thermo-hydraulic performance of a system is commonly evaluated with the help of its various characteristics of temperature and velocity fields. In any fluid and thermal system, the solution of the conservation of the mass, the momentum and the energy equations will pave the way to understand the behavior of parametric fields. Assuming the flow to be laminar, to obtain the velocity and temperature distributions of the system (Figure. 1b), the following general forms of conservation equations are solved [49]. Mass: Momentum:

Here, the generation of volumetric heat and the radiative mode of heat transfer are neglected. Ignoring the thickness of the bottom plate, the Sy -plane of the computational domain is supplied with heat flux of |qv y | (Figure. 1b). The top plate (Sy’ - plane) is considered to be insulated. Water enters the domain through the Sx -plane with

normal velocity Vin and isothermal temperature Tin. After interacting with the VG, the bottom and the top plate, the fluid leaves through the outlet (Sx’ -plane) at the atmospheric condition of Patm and Tatm. Therefore, a forced convection scenario is achieved over the base plate. The flowing fluid is considered to experience a noslip condition due to the top and the bottom plates. The Sz and the Sz’ –planes of the computational domain are considered symmetry for both the velocity and temperature field.

Figure 1. Schematic diagram of (a) physical domain with an array of RVGs fitted over a baseplate, and (b) the computational domain, (c) the RVG and (d) the MRVG In the present work, the governing equations (Eqs. 1-3) are solved using the FVM, with the help of the above-mentioned conditions at the boundary. The ANSYS Fluent 19.2, a commercially available FVM solver is used. Initially, the solver, the governing equations, the boundary conditions, and the considered assumptions are validated using the results of available literature. A grid dependency test is performed prior to the validation work. With the knowledge of u, v, w, and T, from the solution of Eqs. 1-3, various thermo-physical parameters, viz., the Nu, co-efficient of friction Cf, drag coefficient Cd, and thermal performance factor (TPF) 𝜂 are evaluated. The Nu represents the strength of the convective thermal current in the system. To compare the thermal performance of the various systems considered in the present work, a surface average value of Nu is analyzed and is represented as [49].

The hydraulic diameter Dh of the system is taken as the height of the computational domain Ly [46]. The coefficient of thermal enhancement h’’ is defined as [49] |qv y | h" = (T - T ) (5) p

The Re at the inlet of the flow is evaluated using [49] Re =

In a fluid flow system, flow losses are important aspects to characterize the system performance. The nature of flow interaction with the contact surfaces is given by the non-dimensional number known as the surface average skin friction coefficient. Mathematically, it is denoted as [49], Cf =

The coefficient of drag, or Cd”, provides valuable information about the resistance encountered by the VG from the incoming flow. Mathematically, coefficient of drag is expressed as [49], Cd" =

A system with desirable enhancement in heat transfer rate in comparison to the reduction in frictional flow losses is characterized by the TPF (𝜂). Considering the case of a simple RVG as the reference system, the ratio of the normalized Nu to Cf yields 𝜂 [47] h=:

where, Nu and C f are the surface averaged Nu and Cf for the reference case with the RVG, whereas Nu” and Cf ” are the equivalent parameters for the MRVGs. ref

3. Results and discussion

The present work aims to analyze flow over MRVG under the effect of thermal perturbations in the system. The work is carried out numerically to solve the governing equations (Eqs. 1-3) of the considered physics. Any computational work requires validation of the considered mathematical model. An RVG of dimension (hp × lp o × tp) 0.02 × 0.04 × 0.002 m is placed at an angle (φ) of 45 over the bottom plate (Lx= 1.5 m, Lz=0.06 m) [50]. The height of the computational domain (Ly) is taken as 0.04 m. Water is considered to enter the channel at 293 K (Tin) with Re = 350. The incoming fluid leaves the channel through the outlet maintained at 293 K (Tout) and atmospheric pressure condition (Patm). The Sy – plane of the domain 2 is supplied with a constant heat flux of 1000 W/m . To validate the numerical model, all parameters are considered with reference to

Abdollahi and Shams [50]. The considered grid is tested to assess its effect on the results prior to the validation of the numerical solver.

4. Conclusions

The presence of a modified VG can significantly affect heat transfer from the surface to a cooling fluid. This work evaluates the effect of geometric modifications of the AP of an RVG, on the thermo-hydraulic performance of the system. The work is carried out numerically. The impact of various geometrical parameters of the VG on the flow and the thermal field is analyzed. The governing equations are solved using an FVM solver, which is validated using the numerical results of Abdollahi and Shams [50]. A grid independent solution from the current solver shows good agreement with existing previous published work by Abdollahi and Shams [50], with a maximum deviation of 3.5%. The study provides a detailed analysis of the structure of the formed vortices and the temperature distributions, due to changes in the VG geometry. The findings have practical implications in various engineering applications. Following are the observations taken from the study: • The interior angles α1 and α2 have significant impact on heat transfer. The most significant enhancement occurs when α1 (= α2) equals 2π/3, resulting in an 11.13% increase in Nu” compared to the conventional RVG. The maximum values of Cf ” and Cd” are observed, when α1 (= α2) is 5π/6. However, η reaches its maximum value, when α₁ = α₂ = 2π/3. At this condition, the maximum increase is 9.9%. • When α₁ = α₂ = 2π/3, the width is increased from 0.1hp to 0.5hp. As a result, Nu” increases by 27.16%, as compared to the RVG. This enhancement is due to the improved vortex dynamics of the HP and the I. • The maximum value of h” is 237.27 W/m²K, when w = 0.5hp and α₁ = α₂ = 2π/3. In this case, Cf ” and Cd” also increase by 11.98% and 14.12%, respectively. It also yields a maximum 𝜂 of 1.224 compared to the RVG. • An AP closer to the base plate degrades the thermal performance, while positioning it in line with the top surface of the PP improves Cf ” by 11.98%. The obtained value of 2 h” for the configuration is 237.27 W/m K. • The Nu” decreases and Cf ” and Cd” increase with increasing ha. The improvement in 𝜂 of 22.4% is achieved with an

MRVG having α1 = α2 = 2π/3, w = 0.5hp, and ha = 0.935hp. The β of the MRVG does not necessarily provide improvement in heat transfer rate. The strength of the vortices reaches its maximum at β = π/2, for 0 < β <π. However, as β increases from π to 7π/4, the strength of the vortices gradually reduces. It results in a reduction in the overall heat transfer rate. Compared to the RVG, the MRVG with β = 3π/4 shows the maximum increase in Nu” by 27.64%. It also shows a 20.58% increase in Cf ”. The maximum value of h” is found 2 to be 238.17 W/m K where β = 3π/4. However, the MRVG configuration with α1 = α2 = 2π/3, w = 0.5hp, ha = 0.935hp, and β = π/2 achieves an overall heat transfer enhancement (𝜂) of 22.4%.

With the detailed investigation of the considered cases, it has been found that an MRVG with an AP configured with α1=α2 = 2π/3, w=0.5hp, ha=0.935hp and β= 3π/4 yields the highest enhancement in Nu” with 27.64%. However, compared to an RVG, the case with α1=α2 = 2π/3, w=0.5hp, ha=0.935hp and β= π, yields maximum rise in Cf ” and Cd” by 28.88% and 173.85%. Overall, a conjugate heat transfer system demands a balance of enhancement in heat transfer rate with minimum flow losses in the system. In the current study, the MRVG configuration with α1=α2 = 2π/3, w=0.5hp, ha=0.935hp and β= π/2, provides the highest increment in thermal performance factor by 22.4%, as compared to RVG. The value of h” for the RVG is 2 186.59 W/m K. Under the current configuration, the value of h” for 2 the MRVG is found to be 237.27 W/m K. The present work has paved the way for improving the thermal and flow performance of a system with MRVG that utilizes a trapezoid AP. The number of considered cases are limited by considering α1=α2. Moreover, the common surface of AP and PP is always placed parallel to the bottom plate. Consideration of different angles, α1, α2, and the inclination angle of the shared face of AP and PP and its analysis are included as a part of the future endeavor.

Originality statement The authors declare that this manuscript is original and has not been published or submitted elsewhere.

Authorship contributions

Funding and/or conflicts of interests/competing interests The authors have not received funding from any agencies and declare no conflict of interest.

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Schematic diagram of (a) physical domain with an array of RVGs fitted over a baseplate, and (b) the computational domain, (c) the RVG and (d) the MRVG

Sectional distribution of z-velocity at various flow downstream distances (xd)

Temperature distribution at various flow downstream distance (xd)

Comparison of (a) Nu” and Cf ” and (b) Cd” of RVG and different MRVGs with angles α1 (= α2) varying between π/6 and 5π/6 Distribution of z-velocity at different sectional planes downstream of the MRVG with angles α1 (= α2) varying from π/6 to 5π/6

Temperature contours at downstream of the MRVG with α1 (= α2) varying from π/6 to 5π/6

Variation of TPF along the flow over the base plate for different MRVGs

Figure 10 Comparison of (a) Nu” and Cf ” and (b) Cd” of RVG and different MRVGs with varying w Figure 11 Distribution of z-velocity at different sectional planes downstream of the MRVG with w varying from 0.1hp to 0.5hp

Figure 12 Temperature contours at downstream of the MRVG with w varying from 0.1hp to 0.5hp Figure 13 Variation of 𝜂 for RVG and MRVGs with w

Figure 14 Comparison of (a) Nu” and Cf ” and (b) Cd” of RVG and different MRVGs with varying ha.

Figure 15 Distribution of z-velocity at different sectional planes downstream of the MRVG with ha varying from 0 to 0.935hp

Figure 16 Formation of longitudinal vortex upstream of the VG for AP located at various heights on PP

Figure 17 Temperature contours at downstream of the MRVG with ha varying from 0 to 0.935hp

Figure 18 Comparison of 𝜂 of RVG and different MRVGs with height ha varying between 0% hp and 93.5% hp

Figure 19 Comparison of (a) Nu” and Cf ” and (b) Cd” of RVG and different MRVGs with β varying between π/4 and 7π/4 Figure 20 Distribution of z-velocity at different sectional planes downstream of the MRVG where 0 < β < π Figure 21 Distribution of z-velocity at different sectional planes downstream of the MRVG where π ≤ β < 2π Figure 22 Temperature contours at downstream of the MRVG where 0 < β < 2π Figure 23 Comparison of 𝜂 of RVG and different MRVGs with varying β

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