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Mixed convection around a circular cylinder in a buoyancy-assisting flow

Mixed convection around a circular cylinder in a buoyancy-assisting flow Curved and Layer. Struct. 2022; 9:81–95 Research Article Hasan Shakir Majdi, Mahmoud A. Mashkour, Laith Jaafer Habeeb*, and Marko Ilic Mixed convection around a circular cylinder in a buoyancy-assisting flow https://doi.org/10.1515/cls-2022-0008 ding behind these bodies causes unfavorable oscillations Received Oct 05, 2021; accepted Nov 29, 2021 in the aero- or hydro-dynamic loads, leading to undesir- able vibrations in the bodies’ structures. Therefore, a huge Abstract: In this paper, the effect of mixed convection on the effort has been assigned in an attempt to reduce or elimi- flow behavior and heat transfer around a circular cylinder nate the effect of vortex shedding. The literature shows that disclosed to a vertically upward laminar air stream is numer- there are various methods to eliminate or control the vortex ically examine. The buoyancy-aided flow is utilized to elim- shedding like suction, blowing, rotating the body, surface inate and control the vortex shedding of the cylinder. The roughness, and heating the cylinder, see for example [1–5]. influence of the Grashof number, 0 ≤ Gr ≤ 6000, the flow Indeed, this phenomenon in mixed convective flow can and thermal patterns, as well as the local and mean Nus- lead to very intricate physical circumstances due to the high selt number, is investigated at a constant Reynolds number interaction amongst the buoyancy effects of free convec- of 100. The unsteady Navier-Stokes’s equations are solved tion, effects of forced flow, and the effects of secondary employing a finite-volume method to simulate numerically flows generated behind the solid bodies. Many works have the velocity and temperature fields in time and space. The discussed this problem. Badr [6] investigated the mixed results showed periodic instability in the flow and thermal convective flow and heat transfer about a horizontal cylin- fields for a range of Grashof number Gr ≤ 1300. Also, there der for contra and parallel flows for Reynolds numbers is critical value of Grashof number for stopping this instabil- less than 40. Chang and Sa [7] also investigated the flow ity and the vortex shedding formed behind the cylinder, by behavior, Nusselt number variations, and drag coefficient the effect of heating. Thus, by increasing Grashof number from a cylinder at Reynolds numbers of 100 and Grashof between 1400 ≤ Gr ≤ 4000, the periodic flow vanishes and number between104 for cooling and 104 for heating. It was converts into steady flow with twin eddies attached to the found that the vortex shedding can be stopped by increas- cylinder from the back. Furthermore, as Grashof number ing Grashof number higher than 1500. In addition, Wang [8] increases behind Gr ≥ 5000, the flow becomes completely studied mixed convection from an isothermal needle con- attached to the cylinder surface without any separation. taining a hot tip for contra and assisting flows. Michaux Keywords: Mixed convection, circular cylinder, vortex shed- and B’elorgey [9] performed some experiments to study ding, adding flows the effect of buoyancy forces of mixed convection on the wake flow behavior behind a circular cylinder the Reynolds numbers of greater than 130. They observed that the forced convection is predominant when Richardson number is 1 Introduction higher than 0.5, while the natural convection becomes the dominant for Richardson number less than 0.5. Varaprasad Vortex shedding behind cylinders is an interesting topic for et al. [10] examined the effect of buoyancy forces on the many researchers. This is due to that forming vortex shed- mixed convective flow over a circular cylinder and tested its impact on Nusselt number distributions, drag coefficient, wake structure, and Strouhal number. They reported that *Corresponding Author: Laith Jaafer Habeeb: Training and the increase in Grashof number could also increase the Workshop Center, University of Technology, Baghdad, Iraq; Email: Laith.J.Habeeb@uotechnology.edu.iq drag coefficient around the cylinder. Hasan Shakir Majdi: Department of Chemical Engineering and Moreover, Kieft et al. [11] investigated the flow wake Petroleum Industries, Al-Mustaqbal University College, Hillah, structure backward a heated cylinder located in a hori- Babil, Iraq zontal crossflow at constant Reynolds number of 75 and Mahmoud A. Mashkour: Mechanical Engineering Department, Richardson number ranging between 0 and 1. They found University of Technology, Baghdad, Iraq that the heating of cylinder produces vortex shedding in the Marko Ilic: Faculty of Mechanical Engineering, University of Nis, Nis, Serbia Open Access. © 2022 H. Shakir Majdi et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 82 | H. Shakir Majdi et al. rear of cylinder with lower and upper vortices have diverse 2 Physical problem and governing strengths. Biswas and Sarkar [12] studied similar case of Kieft et al. [11] but for range of Reynolds numbers of 10-45. equations Interestingly, the results revealed that heat the cylinder and producing buoyancy forces trigger the vortex shedding to Figure 1 shows the schematic diagram for the geometry of be occurred at low Reynolds number. For example, it was the considered problem in conjunction with the system co- found that at Richardson number of 1.4, the vortex shedding ordinates. Briefly, the problem composes of an unbounded takes place at low Reynolds number of 10, and by increasing circular cylinder of diameter D heated isothermally at a Richardson number to value of 2, the vortex shedding hap- fixed temperature T and subjected to an upward vertical pens at Reynolds number of 45. Sharma et al. [13] studied air stream of velocity v and temperature T . The gravity o o mixed convective flow across a square cylinder for Richard- acts downward, while the buoyancy forces generated due son 0-1 and Reynolds number 1-40. Their investigation did to heating the cylinder act upward. not notice any generation for a vortex shedding behind the cylinder for these ranges of Richardson and Reynolds numbers; however, a consistently steady flow was shown without any shedding. Gandikota et al. [14] examined mixed convective flow around a circular cylinder placed inside a vertical plate channel for two cooling and heating cases at Richardson’s numbers of −0.5 and 0.5, respectively, for a Reynolds number ranging between 50 and 150. Their results indicated using a channel with a higher blockage ratio cre- ates a considerable delay in the separation phenomena of the hydrodynamic boundary layer on the cylinder surface. Guill’en et al. [15] performed an experimental investigation about unsteady mixed convective flow over an isothermal cylinder in a contra-flow situation inside a vertical plate duct at Reynolds number of 170. They realized the influence Figure 1: The physical problem under consideration of duct wall on the formation of flow wake in the back of cylinder. Moreover, Sanyal and Dhiman [16] tested the in- teractions amongst the wakes formed behind two square The standard mass, momentum, and energy equations cylinders under the effect of mixed convection for Richard- described below in the non-dimensional form, are used to son number between and Reynolds number between 1-40, calculate the flow and temperature fields around the cylin- and for heavy liquid that possesses Prandtl number of 50. der, and then the amount of heat released from the cylinder, Chatterjee and Mondal [17–22] have made many studies for a transient, laminar, incompressible, two-dimensional about the effects of buoyancy forces on heat transfer and mixed convective flow as follows: flow behavior about a cylinder. ∂u ∂v + = 0, (1) The current literature highlighted that the increase ∂x ∂y in Grashof number might provoke an elimination for the (︂ )︂ vortex shedding; however, the influences of the buoyancy (︁ )︁ ∂u ∂u ∂u 1 ∂p forces on eliminating the vortex shedding behind a cylinder + u + v = ∇ u − , (2) ∂t ∂x ∂y Re ∂x in an assisting flow has been investigated yet. Therefore, the current study aims to test the buoyancy effects of mixed (︂ )︂ (︁ )︁ ∂v ∂v ∂v 1 ∂p convection from a circular isothermal cylinder positioned + u + v = ∇ v − + RiT, (3) ∂t ∂x ∂y Re ∂y in an upward vertical stream on the flow behavior and heat transfer characteristics. In this paper, we present the phys- (︂ )︂ (︁ )︁ ical problem and the equations that govern the flow and ∂T ∂T ∂T 1 + u + v = ∇ T , (4) thermal fields. Then, the numerical method that was used ∂t ∂x ∂y Re · Pr to solve the governing equations is briefly described. After where u and v are the non-dimensional velocities in hor- that, the results and their discussion are outlined. Lastly, izontal and vertical orientations, respectively; T, p and t we draw the study’s findings in the conclusion section. are the non-dimensional temperature, pressure, and time, Mixed convection around a circular cylinder in a buoyancy-assisting flow | 83 respectively. We used the following dimensional param- An interpolation method based on a body-force-weighted eters for obtaining the aforementioned non-dimensional pressure technique is utilized for interpolating the interface equations: pressure from the cell center value. The discretized equa- tions are then numerically solved by an in-house solver. In ′ ′ ′ ′ ′ u v x y p u = , v = , x = , y = , p = , the current computations, we used non-uniform computa- v v D D o o ρv tional mesh with fine clustering distributions in the vicinity ′ ′ t T − T of the cylinder surface walls for accurate calculation for t = , T = v D T − T o o the bigger variables’ gradients. Figure 2 displays the com- The principal dimensionless groups that control this kind putational mesh used for the whole computational domain. of flow are Reynolds number Re, Richardson number Ri, and Prandtl Pr, which are defined as follows: v ρD Gr c µ o p Re = , Ri = , Pr = , (5) µ Re k where, c , µ and ρ are the uid fl heat capacity, the uid fl dynamic viscosity, and the uid fl density. Gr is the Grashof number as follows: 2 3 ρ · g · βD (T − T ) Gr = , (6) where, k is the uid fl thermal conductivity, and g the accel- eration due to the earth gravity. Local Nusselt number Nu represents the local heat transfer from the hot surface around the cylinder, and is calculated as: hD ∂T Nu = = − , (7) k ∂n where, h is the local convective heat transfer coefficient around the cylinder perimeter, and n is the direction per- Figure 2: (Left) The computational grid for the entire domain, (Right) pendicular to the cylinder wall. Then, this quantity is in- Zooming for the mesh around the cylinder tegrated over the cylinder surface to determine mean heat transfer Nu. The boundary conditions that are employed in the present study can be expressed as: 3 Numerical method u = 0, v = 1, T = 0, at Inlet (8) o o o ∂u ∂v ∂T = 0, = 0, = 0, at Outlet The aforementioned governing Eqs. (1)–(4) along with ∂y ∂y ∂y the boundary conditions elaborated in Eq. (8) below, are ∂u ∂v ∂T = 0, = 0, = 0, at Artificial solved numerically employing a finite-volume method ∂x ∂x ∂x based on the SIMPLEC algorithm described in (Doormal vertical borders and Raithby [23]). In this method, the temporal discretiza- u = v = 0, T = 1, at cylinder wall tion is conducted by a second order implicit Adams- Bash- (0 < θ < 360). forth scheme. Whereas, a second order upwind scheme is used for discretizing the convective terms, and a cen- A mesh sensitivity study is performed to choose the best tral difference scheme is used for discretizing the diffusive economic grid size. The numerical solution was checked for terms, in both the momentum and energy equations. The a mesh independency using different non-uniform mesh Courant-Friedrichs-Lewy and the grid Fourier number cri- sizes S1, S2, S3, S4, S5, and S6, as shown in Table 1. This teria are employed to avoid any computational oscillations. study was performed for different pertinent parameters. It For convergence, the time interval was taken to be changed was found that the mesh size S3 of (120×298) along x and y between 0.001 and 0.01 to calculate an optimal value for directions, respectively, can be used with an error less than less numerical time, but with satisfactorily precise results. 0.1%. 84 | H. Shakir Majdi et al. Table 1: Mesh sizes employed to examine the mesh independency called wake attached to the cylinder back, as shown in Fig- ure 6. The breakdown of the Van Karman vortex street hap- Mesh (∆x × ∆y ) pens probably by the reason of that the points of the flow S1 (92 × 235) separation on the right and left sides of the cylinder wall S2 (102 × 255) move downstream by the impact of the buoyancy forces as S3 (120 × 298) Grashof number increases. Moving of the separation loca- S4 (136 × 340) tions downstream the cylinder wall might raise from the in- S5 (150 × 375) teractions between the eddies; consequently, vanishing of the Van Karman vortex street takes place and the flow turns completely into symmetric steady flow with stationary twin vortices behind the cylinder. It can also see that these twin eddies shrink considerably as Grashof number increases from 1300 to 4000, displayed in Figure 8. Furthermore, by increasing Grashof number to more than 4000, the results indicate that the wake flow vanishes and disappear, and the flow becomes entirely attached to the cylinder surface without any separation. Therefore, it can be obviously con- cluded that the thermal buoyancy forces can significantly affect the flow behavior behind the cylinder; thus, they can control or eliminate forming the vortex shedding, as well as reduce or eliminate forming the wake flow. Indeed, this phenomenon may decrease the drag forces on the cylinder Figure 3: Verification for the numerical algorithm employed in our exposed to the vertical flow stream. code Moreover, the g fi ures also show how the changing in the flow behavior the rear of the cylinder affect the shape of the thermal plume in this region, also the thickness of the The accuracy of our numerical program was verified by thermal boundary layer formed on the cylinder wall. There- comparing its results with experimental results previously fore, this can change the heat transfer from the cylinder as published Nasr et al. [24] for the problem of forced convec- it will be seen later. tion about a cylinder of 12.7 mm diameter, and embedded Another important objective of the current study is to in- in porous medium, which compromises from aluminum vestigate energy exchange from the cylinder to the adjacent spheres of 12.23 mm diameter. Fixed porosity medium of flowing uid. fl For this purpose, the variations of transient 0.37 with thermal conductivity ratio of 8.7 were used. This mean Nusselt number are shown in Figure 10 for Grashof comparison is demonstrated in Figure 3, and depicts a good numbers of 0 and 1300 when the vortex shedding exists, matching between the experimental results and the numer- and in Figure 11 for Grashof number more than 1400, when ical results of our code. the flow becomes completely steady flow. In Figure 10, one can see that at Gr = 0 and 1300, the behavior of the mean Nusselt number with the time after a long transient period 4 Results and discussion becomes fully periodic. Also, this transient period increases significantly as Grashof number increase from 0 to 1300. In the present study, the numerical results were obtained In addition, importantly the increase in Grashof number for Prandtl number Pr = 0.71 for air as working uid, fl and leads to a slight increase in the mean Nusselt number. How- at constant Reynolds number of Re = 100. Figures 4–9 show ever, in Figure 11, the results show that the behavior of the the vorticity, streamlines, and isotherms patterns around mean Nusselt number with the time after a short transient the heated cylinder for different Grashof numbers Gr = 0– period becomes fully steady for Gr ≥ 1400. Also, the tran- 6000. It is shown in Figure 4 that at Gr = 0 and 1300 the sient period is shown to be decreased as Grashof number mixed convective flow over the cylinder body generates a increases. Surprisingly, the results reveal that for the in- periodic secondary flow, which is called Van Karman vortex crease in Grashof number from 1400 to 4000, the amount street. Also, as Grashof number increases, the van Karman of mean Nusselt number decreases slightly as Grashof num- vortex street at the cylinder back vanishes, and the flow ber increases. This is when the twin vortices (wake flow) converts completely to a steady flow with double eddies behind the cylinder exist. But, for the increase in Grashof Mixed convection around a circular cylinder in a buoyancy-assisting flow | 85 (a) vorticity (b) stream lines Gr = 0 Gr = 1300 Gr = 0 Gr = 1300 Figure 4: (a) Vorticity patterns, and (b) streamlines, for two Grashof numbers Gr = 0.0 and 1300, at Re = 100 86 | H. Shakir Majdi et al. Gr = 0 Gr = 1300 Figure 5: Isotherms patterns for two Grashof numbers Gr = 0.0 and 1300, at Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 87 (a) vorticity (b) stream lines Gr = 1400 Gr = 4000 Gr = 1400 Gr = 4000 Figure 6: (a) Vorticity patterns, and (b) streamlines, for two Grashof numbers Gr = 1400 and 4000, at Re = 100 88 | H. Shakir Majdi et al. Gr = 1400 Gr = 4000 Figure 7: Isotherms patterns for two Grashof numbers Gr = 1400 and 4000, at Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 89 (a) vorticity (b) stream lines Gr = 5000 Gr = 6000 Gr = 5000 Gr = 6000 Figure 8: (a) Vorticity patterns, and (b) streamlines, for two Grashof numbers Gr = 5000 and 6000, at Re = 100 90 | H. Shakir Majdi et al. Gr = 5000 Gr = 6000 Figure 9: Isotherms patterns for two Grashof numbers Gr = 5000 and 6000, at Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 91 number from 5000 to 6000, when these twin vortices disap- pear and the flow becomes completely attached along the cylinder surface, the effect of Grashof number on mean Nus- selt number alters. Thus, it is seen that the mean Nusselt number starts increasing as Grashof number increases. For best understanding to the flow characteristics un- der the instability situation, a series of images for the in- stantaneous streamlines, vorticity, and isotherms patterns, are shown in Figures 12-15 over full periodic cycles of oscil- lation in Nusselt number, which, which corresponds to full vortex shedding period at Gr = 0 and 1300, respectively. It is seen that in this instability, a couple of eddies is developed, and is shed alternately in the downstream direction. The g fi ures show that for both cases of Grashof numbers, simi- lar mode of periodicity is produced, with a slight increase in the oscillation amplitude and frequency is observed at Gr = 1300. The images of the vorticity depict the formation and shedding of the positive (blue) and negative (red) vor- tices in the wake region during the shedding period. The images also depict the development of the thermal wake associated with the hydro-dynamic wake at the same cycle times. It is shown that the thermal boundary layer after the Figure 10: Transient behavior of Nusselt number with the time, cylinder splits and transfers within the wake zone, and the Grashof numbers Gr = 0.0 and 1300, at Re = 100 hot plumes emerging in the places of the wake eddies. In fact, as the wake eddies are rolling up and shedding peri- odically behind the cylinder, as the thermal boundary layer on the cylinder wall changes producing a clear uctuation fl in Nusselt number with the time, as demonstrated in the upper plots of each g fi ure. The distribution of local Nusselt number Nu along the cylinder circumference for a polar angle between θ = 0−360 is illustrated in Figure 16 for different Grashof num- bers Gr = 0–5000, at Re = 100. The positions of θ = 90 and 270 represent the rear and the front stagnation points in relation to the incoming flow air. It can be seen that as Grashof number increases, the local Nusselt number increases minimally over the surface close to the front sep- aration point owing to the huge uid fl acceleration, which reduces significantly the thickness of the boundary layer in this area of the cylinder. At low Grashof number Gr ≤ 1400, the minimum local Nusselt number occurs at the sides of ∘ ∘ ∘ the cylinders of θ ≈ 40 and ≈ 140 . However, as Grashof increases behind Gr ≥ 4000, the local Nusselt number de- crease in the back area between , until the minimum local Nusselt number occurs at the rare separation point of the cylinder at θ ≈ 90 . This is attributable to the heating effect, which suppresses the vortex shedding, delays the Figure 11: Transient behavior of Nusselt number with the time, flow separation, and shifts the points of separation from Grashof numbers Gr = 1400, 4000, 5000 and 6000, at Re = 100 the cylinder sides into the rare point until the complete flow attachment is happened around the cylinder, and the flow separation is vanished. 92 | H. Shakir Majdi et al. t t t t t 1 2 3 4 5 Figure 12: Stream patterns over a half-full transient period at Grashof number Gr = 0.0 and Re = 100 (a) vorticity (b) Isotherms Figure 13: (a) Vorticity, and (b) isotherms, patterns over a half-full transient period at Grashof number Gr = 0.0 and Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 93 t t t t t 1 2 3 4 5 Figure 14: Stream patterns over a half-full transient period at Grashof number Gr = 1300 and Re = 100 (a) vorticity (b) Isotherms Figure 15: (a) Vorticity, and (b) isotherms, patterns over a half-full transient period at Grashof number Gr = 1300 and Re = 100 94 | H. Shakir Majdi et al. Funding information: The authors state no funding in- volved. Author contributions: All authors have accepted responsi- bility for the entire content of this manuscript and approved its submission. Conflict of interest: The authors state no conflict of inter- est. 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Mixed convection around a circular cylinder in a buoyancy-assisting flow

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Curved and Layer. Struct. 2022; 9:81–95 Research Article Hasan Shakir Majdi, Mahmoud A. Mashkour, Laith Jaafer Habeeb*, and Marko Ilic Mixed convection around a circular cylinder in a buoyancy-assisting flow https://doi.org/10.1515/cls-2022-0008 ding behind these bodies causes unfavorable oscillations Received Oct 05, 2021; accepted Nov 29, 2021 in the aero- or hydro-dynamic loads, leading to undesir- able vibrations in the bodies’ structures. Therefore, a huge Abstract: In this paper, the effect of mixed convection on the effort has been assigned in an attempt to reduce or elimi- flow behavior and heat transfer around a circular cylinder nate the effect of vortex shedding. The literature shows that disclosed to a vertically upward laminar air stream is numer- there are various methods to eliminate or control the vortex ically examine. The buoyancy-aided flow is utilized to elim- shedding like suction, blowing, rotating the body, surface inate and control the vortex shedding of the cylinder. The roughness, and heating the cylinder, see for example [1–5]. influence of the Grashof number, 0 ≤ Gr ≤ 6000, the flow Indeed, this phenomenon in mixed convective flow can and thermal patterns, as well as the local and mean Nus- lead to very intricate physical circumstances due to the high selt number, is investigated at a constant Reynolds number interaction amongst the buoyancy effects of free convec- of 100. The unsteady Navier-Stokes’s equations are solved tion, effects of forced flow, and the effects of secondary employing a finite-volume method to simulate numerically flows generated behind the solid bodies. Many works have the velocity and temperature fields in time and space. The discussed this problem. Badr [6] investigated the mixed results showed periodic instability in the flow and thermal convective flow and heat transfer about a horizontal cylin- fields for a range of Grashof number Gr ≤ 1300. Also, there der for contra and parallel flows for Reynolds numbers is critical value of Grashof number for stopping this instabil- less than 40. Chang and Sa [7] also investigated the flow ity and the vortex shedding formed behind the cylinder, by behavior, Nusselt number variations, and drag coefficient the effect of heating. Thus, by increasing Grashof number from a cylinder at Reynolds numbers of 100 and Grashof between 1400 ≤ Gr ≤ 4000, the periodic flow vanishes and number between104 for cooling and 104 for heating. It was converts into steady flow with twin eddies attached to the found that the vortex shedding can be stopped by increas- cylinder from the back. Furthermore, as Grashof number ing Grashof number higher than 1500. In addition, Wang [8] increases behind Gr ≥ 5000, the flow becomes completely studied mixed convection from an isothermal needle con- attached to the cylinder surface without any separation. taining a hot tip for contra and assisting flows. Michaux Keywords: Mixed convection, circular cylinder, vortex shed- and B’elorgey [9] performed some experiments to study ding, adding flows the effect of buoyancy forces of mixed convection on the wake flow behavior behind a circular cylinder the Reynolds numbers of greater than 130. They observed that the forced convection is predominant when Richardson number is 1 Introduction higher than 0.5, while the natural convection becomes the dominant for Richardson number less than 0.5. Varaprasad Vortex shedding behind cylinders is an interesting topic for et al. [10] examined the effect of buoyancy forces on the many researchers. This is due to that forming vortex shed- mixed convective flow over a circular cylinder and tested its impact on Nusselt number distributions, drag coefficient, wake structure, and Strouhal number. They reported that *Corresponding Author: Laith Jaafer Habeeb: Training and the increase in Grashof number could also increase the Workshop Center, University of Technology, Baghdad, Iraq; Email: Laith.J.Habeeb@uotechnology.edu.iq drag coefficient around the cylinder. Hasan Shakir Majdi: Department of Chemical Engineering and Moreover, Kieft et al. [11] investigated the flow wake Petroleum Industries, Al-Mustaqbal University College, Hillah, structure backward a heated cylinder located in a hori- Babil, Iraq zontal crossflow at constant Reynolds number of 75 and Mahmoud A. Mashkour: Mechanical Engineering Department, Richardson number ranging between 0 and 1. They found University of Technology, Baghdad, Iraq that the heating of cylinder produces vortex shedding in the Marko Ilic: Faculty of Mechanical Engineering, University of Nis, Nis, Serbia Open Access. © 2022 H. Shakir Majdi et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License 82 | H. Shakir Majdi et al. rear of cylinder with lower and upper vortices have diverse 2 Physical problem and governing strengths. Biswas and Sarkar [12] studied similar case of Kieft et al. [11] but for range of Reynolds numbers of 10-45. equations Interestingly, the results revealed that heat the cylinder and producing buoyancy forces trigger the vortex shedding to Figure 1 shows the schematic diagram for the geometry of be occurred at low Reynolds number. For example, it was the considered problem in conjunction with the system co- found that at Richardson number of 1.4, the vortex shedding ordinates. Briefly, the problem composes of an unbounded takes place at low Reynolds number of 10, and by increasing circular cylinder of diameter D heated isothermally at a Richardson number to value of 2, the vortex shedding hap- fixed temperature T and subjected to an upward vertical pens at Reynolds number of 45. Sharma et al. [13] studied air stream of velocity v and temperature T . The gravity o o mixed convective flow across a square cylinder for Richard- acts downward, while the buoyancy forces generated due son 0-1 and Reynolds number 1-40. Their investigation did to heating the cylinder act upward. not notice any generation for a vortex shedding behind the cylinder for these ranges of Richardson and Reynolds numbers; however, a consistently steady flow was shown without any shedding. Gandikota et al. [14] examined mixed convective flow around a circular cylinder placed inside a vertical plate channel for two cooling and heating cases at Richardson’s numbers of −0.5 and 0.5, respectively, for a Reynolds number ranging between 50 and 150. Their results indicated using a channel with a higher blockage ratio cre- ates a considerable delay in the separation phenomena of the hydrodynamic boundary layer on the cylinder surface. Guill’en et al. [15] performed an experimental investigation about unsteady mixed convective flow over an isothermal cylinder in a contra-flow situation inside a vertical plate duct at Reynolds number of 170. They realized the influence Figure 1: The physical problem under consideration of duct wall on the formation of flow wake in the back of cylinder. Moreover, Sanyal and Dhiman [16] tested the in- teractions amongst the wakes formed behind two square The standard mass, momentum, and energy equations cylinders under the effect of mixed convection for Richard- described below in the non-dimensional form, are used to son number between and Reynolds number between 1-40, calculate the flow and temperature fields around the cylin- and for heavy liquid that possesses Prandtl number of 50. der, and then the amount of heat released from the cylinder, Chatterjee and Mondal [17–22] have made many studies for a transient, laminar, incompressible, two-dimensional about the effects of buoyancy forces on heat transfer and mixed convective flow as follows: flow behavior about a cylinder. ∂u ∂v + = 0, (1) The current literature highlighted that the increase ∂x ∂y in Grashof number might provoke an elimination for the (︂ )︂ vortex shedding; however, the influences of the buoyancy (︁ )︁ ∂u ∂u ∂u 1 ∂p forces on eliminating the vortex shedding behind a cylinder + u + v = ∇ u − , (2) ∂t ∂x ∂y Re ∂x in an assisting flow has been investigated yet. Therefore, the current study aims to test the buoyancy effects of mixed (︂ )︂ (︁ )︁ ∂v ∂v ∂v 1 ∂p convection from a circular isothermal cylinder positioned + u + v = ∇ v − + RiT, (3) ∂t ∂x ∂y Re ∂y in an upward vertical stream on the flow behavior and heat transfer characteristics. In this paper, we present the phys- (︂ )︂ (︁ )︁ ical problem and the equations that govern the flow and ∂T ∂T ∂T 1 + u + v = ∇ T , (4) thermal fields. Then, the numerical method that was used ∂t ∂x ∂y Re · Pr to solve the governing equations is briefly described. After where u and v are the non-dimensional velocities in hor- that, the results and their discussion are outlined. Lastly, izontal and vertical orientations, respectively; T, p and t we draw the study’s findings in the conclusion section. are the non-dimensional temperature, pressure, and time, Mixed convection around a circular cylinder in a buoyancy-assisting flow | 83 respectively. We used the following dimensional param- An interpolation method based on a body-force-weighted eters for obtaining the aforementioned non-dimensional pressure technique is utilized for interpolating the interface equations: pressure from the cell center value. The discretized equa- tions are then numerically solved by an in-house solver. In ′ ′ ′ ′ ′ u v x y p u = , v = , x = , y = , p = , the current computations, we used non-uniform computa- v v D D o o ρv tional mesh with fine clustering distributions in the vicinity ′ ′ t T − T of the cylinder surface walls for accurate calculation for t = , T = v D T − T o o the bigger variables’ gradients. Figure 2 displays the com- The principal dimensionless groups that control this kind putational mesh used for the whole computational domain. of flow are Reynolds number Re, Richardson number Ri, and Prandtl Pr, which are defined as follows: v ρD Gr c µ o p Re = , Ri = , Pr = , (5) µ Re k where, c , µ and ρ are the uid fl heat capacity, the uid fl dynamic viscosity, and the uid fl density. Gr is the Grashof number as follows: 2 3 ρ · g · βD (T − T ) Gr = , (6) where, k is the uid fl thermal conductivity, and g the accel- eration due to the earth gravity. Local Nusselt number Nu represents the local heat transfer from the hot surface around the cylinder, and is calculated as: hD ∂T Nu = = − , (7) k ∂n where, h is the local convective heat transfer coefficient around the cylinder perimeter, and n is the direction per- Figure 2: (Left) The computational grid for the entire domain, (Right) pendicular to the cylinder wall. Then, this quantity is in- Zooming for the mesh around the cylinder tegrated over the cylinder surface to determine mean heat transfer Nu. The boundary conditions that are employed in the present study can be expressed as: 3 Numerical method u = 0, v = 1, T = 0, at Inlet (8) o o o ∂u ∂v ∂T = 0, = 0, = 0, at Outlet The aforementioned governing Eqs. (1)–(4) along with ∂y ∂y ∂y the boundary conditions elaborated in Eq. (8) below, are ∂u ∂v ∂T = 0, = 0, = 0, at Artificial solved numerically employing a finite-volume method ∂x ∂x ∂x based on the SIMPLEC algorithm described in (Doormal vertical borders and Raithby [23]). In this method, the temporal discretiza- u = v = 0, T = 1, at cylinder wall tion is conducted by a second order implicit Adams- Bash- (0 < θ < 360). forth scheme. Whereas, a second order upwind scheme is used for discretizing the convective terms, and a cen- A mesh sensitivity study is performed to choose the best tral difference scheme is used for discretizing the diffusive economic grid size. The numerical solution was checked for terms, in both the momentum and energy equations. The a mesh independency using different non-uniform mesh Courant-Friedrichs-Lewy and the grid Fourier number cri- sizes S1, S2, S3, S4, S5, and S6, as shown in Table 1. This teria are employed to avoid any computational oscillations. study was performed for different pertinent parameters. It For convergence, the time interval was taken to be changed was found that the mesh size S3 of (120×298) along x and y between 0.001 and 0.01 to calculate an optimal value for directions, respectively, can be used with an error less than less numerical time, but with satisfactorily precise results. 0.1%. 84 | H. Shakir Majdi et al. Table 1: Mesh sizes employed to examine the mesh independency called wake attached to the cylinder back, as shown in Fig- ure 6. The breakdown of the Van Karman vortex street hap- Mesh (∆x × ∆y ) pens probably by the reason of that the points of the flow S1 (92 × 235) separation on the right and left sides of the cylinder wall S2 (102 × 255) move downstream by the impact of the buoyancy forces as S3 (120 × 298) Grashof number increases. Moving of the separation loca- S4 (136 × 340) tions downstream the cylinder wall might raise from the in- S5 (150 × 375) teractions between the eddies; consequently, vanishing of the Van Karman vortex street takes place and the flow turns completely into symmetric steady flow with stationary twin vortices behind the cylinder. It can also see that these twin eddies shrink considerably as Grashof number increases from 1300 to 4000, displayed in Figure 8. Furthermore, by increasing Grashof number to more than 4000, the results indicate that the wake flow vanishes and disappear, and the flow becomes entirely attached to the cylinder surface without any separation. Therefore, it can be obviously con- cluded that the thermal buoyancy forces can significantly affect the flow behavior behind the cylinder; thus, they can control or eliminate forming the vortex shedding, as well as reduce or eliminate forming the wake flow. Indeed, this phenomenon may decrease the drag forces on the cylinder Figure 3: Verification for the numerical algorithm employed in our exposed to the vertical flow stream. code Moreover, the g fi ures also show how the changing in the flow behavior the rear of the cylinder affect the shape of the thermal plume in this region, also the thickness of the The accuracy of our numerical program was verified by thermal boundary layer formed on the cylinder wall. There- comparing its results with experimental results previously fore, this can change the heat transfer from the cylinder as published Nasr et al. [24] for the problem of forced convec- it will be seen later. tion about a cylinder of 12.7 mm diameter, and embedded Another important objective of the current study is to in- in porous medium, which compromises from aluminum vestigate energy exchange from the cylinder to the adjacent spheres of 12.23 mm diameter. Fixed porosity medium of flowing uid. fl For this purpose, the variations of transient 0.37 with thermal conductivity ratio of 8.7 were used. This mean Nusselt number are shown in Figure 10 for Grashof comparison is demonstrated in Figure 3, and depicts a good numbers of 0 and 1300 when the vortex shedding exists, matching between the experimental results and the numer- and in Figure 11 for Grashof number more than 1400, when ical results of our code. the flow becomes completely steady flow. In Figure 10, one can see that at Gr = 0 and 1300, the behavior of the mean Nusselt number with the time after a long transient period 4 Results and discussion becomes fully periodic. Also, this transient period increases significantly as Grashof number increase from 0 to 1300. In the present study, the numerical results were obtained In addition, importantly the increase in Grashof number for Prandtl number Pr = 0.71 for air as working uid, fl and leads to a slight increase in the mean Nusselt number. How- at constant Reynolds number of Re = 100. Figures 4–9 show ever, in Figure 11, the results show that the behavior of the the vorticity, streamlines, and isotherms patterns around mean Nusselt number with the time after a short transient the heated cylinder for different Grashof numbers Gr = 0– period becomes fully steady for Gr ≥ 1400. Also, the tran- 6000. It is shown in Figure 4 that at Gr = 0 and 1300 the sient period is shown to be decreased as Grashof number mixed convective flow over the cylinder body generates a increases. Surprisingly, the results reveal that for the in- periodic secondary flow, which is called Van Karman vortex crease in Grashof number from 1400 to 4000, the amount street. Also, as Grashof number increases, the van Karman of mean Nusselt number decreases slightly as Grashof num- vortex street at the cylinder back vanishes, and the flow ber increases. This is when the twin vortices (wake flow) converts completely to a steady flow with double eddies behind the cylinder exist. But, for the increase in Grashof Mixed convection around a circular cylinder in a buoyancy-assisting flow | 85 (a) vorticity (b) stream lines Gr = 0 Gr = 1300 Gr = 0 Gr = 1300 Figure 4: (a) Vorticity patterns, and (b) streamlines, for two Grashof numbers Gr = 0.0 and 1300, at Re = 100 86 | H. Shakir Majdi et al. Gr = 0 Gr = 1300 Figure 5: Isotherms patterns for two Grashof numbers Gr = 0.0 and 1300, at Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 87 (a) vorticity (b) stream lines Gr = 1400 Gr = 4000 Gr = 1400 Gr = 4000 Figure 6: (a) Vorticity patterns, and (b) streamlines, for two Grashof numbers Gr = 1400 and 4000, at Re = 100 88 | H. Shakir Majdi et al. Gr = 1400 Gr = 4000 Figure 7: Isotherms patterns for two Grashof numbers Gr = 1400 and 4000, at Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 89 (a) vorticity (b) stream lines Gr = 5000 Gr = 6000 Gr = 5000 Gr = 6000 Figure 8: (a) Vorticity patterns, and (b) streamlines, for two Grashof numbers Gr = 5000 and 6000, at Re = 100 90 | H. Shakir Majdi et al. Gr = 5000 Gr = 6000 Figure 9: Isotherms patterns for two Grashof numbers Gr = 5000 and 6000, at Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 91 number from 5000 to 6000, when these twin vortices disap- pear and the flow becomes completely attached along the cylinder surface, the effect of Grashof number on mean Nus- selt number alters. Thus, it is seen that the mean Nusselt number starts increasing as Grashof number increases. For best understanding to the flow characteristics un- der the instability situation, a series of images for the in- stantaneous streamlines, vorticity, and isotherms patterns, are shown in Figures 12-15 over full periodic cycles of oscil- lation in Nusselt number, which, which corresponds to full vortex shedding period at Gr = 0 and 1300, respectively. It is seen that in this instability, a couple of eddies is developed, and is shed alternately in the downstream direction. The g fi ures show that for both cases of Grashof numbers, simi- lar mode of periodicity is produced, with a slight increase in the oscillation amplitude and frequency is observed at Gr = 1300. The images of the vorticity depict the formation and shedding of the positive (blue) and negative (red) vor- tices in the wake region during the shedding period. The images also depict the development of the thermal wake associated with the hydro-dynamic wake at the same cycle times. It is shown that the thermal boundary layer after the Figure 10: Transient behavior of Nusselt number with the time, cylinder splits and transfers within the wake zone, and the Grashof numbers Gr = 0.0 and 1300, at Re = 100 hot plumes emerging in the places of the wake eddies. In fact, as the wake eddies are rolling up and shedding peri- odically behind the cylinder, as the thermal boundary layer on the cylinder wall changes producing a clear uctuation fl in Nusselt number with the time, as demonstrated in the upper plots of each g fi ure. The distribution of local Nusselt number Nu along the cylinder circumference for a polar angle between θ = 0−360 is illustrated in Figure 16 for different Grashof num- bers Gr = 0–5000, at Re = 100. The positions of θ = 90 and 270 represent the rear and the front stagnation points in relation to the incoming flow air. It can be seen that as Grashof number increases, the local Nusselt number increases minimally over the surface close to the front sep- aration point owing to the huge uid fl acceleration, which reduces significantly the thickness of the boundary layer in this area of the cylinder. At low Grashof number Gr ≤ 1400, the minimum local Nusselt number occurs at the sides of ∘ ∘ ∘ the cylinders of θ ≈ 40 and ≈ 140 . However, as Grashof increases behind Gr ≥ 4000, the local Nusselt number de- crease in the back area between , until the minimum local Nusselt number occurs at the rare separation point of the cylinder at θ ≈ 90 . This is attributable to the heating effect, which suppresses the vortex shedding, delays the Figure 11: Transient behavior of Nusselt number with the time, flow separation, and shifts the points of separation from Grashof numbers Gr = 1400, 4000, 5000 and 6000, at Re = 100 the cylinder sides into the rare point until the complete flow attachment is happened around the cylinder, and the flow separation is vanished. 92 | H. Shakir Majdi et al. t t t t t 1 2 3 4 5 Figure 12: Stream patterns over a half-full transient period at Grashof number Gr = 0.0 and Re = 100 (a) vorticity (b) Isotherms Figure 13: (a) Vorticity, and (b) isotherms, patterns over a half-full transient period at Grashof number Gr = 0.0 and Re = 100 Mixed convection around a circular cylinder in a buoyancy-assisting flow | 93 t t t t t 1 2 3 4 5 Figure 14: Stream patterns over a half-full transient period at Grashof number Gr = 1300 and Re = 100 (a) vorticity (b) Isotherms Figure 15: (a) Vorticity, and (b) isotherms, patterns over a half-full transient period at Grashof number Gr = 1300 and Re = 100 94 | H. Shakir Majdi et al. Funding information: The authors state no funding in- volved. Author contributions: All authors have accepted responsi- bility for the entire content of this manuscript and approved its submission. Conflict of interest: The authors state no conflict of inter- est. 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Journal

Curved and Layered Structuresde Gruyter

Published: Jan 1, 2022

Keywords: Mixed convection; circular cylinder; vortex shedding; adding flows

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