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Three dimensional heat transfer from a square cylinder at low Reynolds numbers
International Journal of Thermal Sciences 119 (2017) 37e50
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Three dimensional heat transfer from a square cylinder at low Reynolds numbers Necati Mahir Eskis¸ehir Osmangazi University, School of Engineering and Architecture, Mechanical Engineering Department, 26480 Batı Mes¸elik, Eskis¸ehir, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 January 2017 Received in revised form 26 April 2017 Accepted 27 April 2017 Available online 1 June 2017
Three-dimensional unsteady flow and heat transfer from an isothermal square cylinder subjected to crossflow of air is numerically investigated. The governing continuity, momentum and energy equations are solved using implicit fractional step method. The first order spatial derivatives are discretized using a third-order upwind scheme while the second order derivatives from the viscous terms are discretized by central-difference formula. The momentum equations are solved separately using Crank-Nicholson timestepping method. The Poisson type equations are solved using Jacobi method with Chebyshev acceleration procedure. The heat transfer characteristics of a square cylinder subjected cross flow of air and the threedimensionality of the flow and its effects are assessed. Three dimensional instantaneous isotemperature surfaces and in the wake region and local Nusselt number variations on the cylinder surfaces along z-axis were provided. The time histories of the Nusselt numbers at the cylinder surfaces are obtained. Also instantaneous and mean deviation of the local Nusselt numbers on the cylinder surfaces both in spanwise direction and x-y plane are provided for Re ¼ 185 and 250 which are representative of typical A-mode and B-mode vortex structures in spanwise direction respectively. © 2017 Elsevier Masson SAS. All rights reserved.
Keywords: 3-D heat transfer Square cylinder Forced convection
1. Introduction The fluid flow and heat transfer around a bluff body has received increasing interest in many engineering fields due to its applications such as heat exchangers, cooling systems, cooling of electronic systems, pipelines, flow around nuclear rods, heat lost from buildings, etc. Therefore, flow around the unconfined circular and/ or square cylinders has been experimentally and numerically studied for the last decades. It has been long established that the fluid flow around a cylinder, at small Reynolds numbers (Re < 165), is laminar and twodimensional. At relatively large Reynolds numbers (Re > 165), the flow becomes three-dimensional having well-known discontinuities at the St-Re curve. This discontinuity was reported by Williamson [1e3], Luo et al. [4] and others. The transition phenomena in the wake of a circular cylinder were studied in detail by Luo et al. [4]. Mode A and mode B type instabilities in the wake of a square cylinder were studied and determined by identifying the discontinuities in the St-Re curves.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijthermalsci.2017.04.031 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.
The authors found that Reynolds number had mean values of 160 and 204 for the onset of the mode A and B type instabilities, respectively. The experimental study of Luo et al. [4] also showed that the aspect ratio of the cylinder and the end plates which used to promote a three-dimensional flow have a strong influence on the wake of a cylinder. They obtained the St-Re discontinuity by resolving the flow for sufficiently small increments of the Reynolds number. It was shown that when the cylinder aspect ratio was less than 22.2, the magnitude of the Strouhal number increases with the cylinder aspect ratio and remains almost constant for larger aspect ratios. Large scale vortex shedding in the wake region and the existence of discontinuities in the St-Re relationships was studied by Williamson [1e3]. It was found that both of these wake characteristics are directly related to each other. The St discontinuities are caused by a transition from one oblique shedding mode to another oblique mode. Vortex dislocation is observed due to vortices in neighboring cells which move out of phase with each other. However, by manipulating the spanwise end conditions of the cylinder, it becomes possible to induce parallel vortex shedding which results in continuous St-Re curve. Luo et al. [5] experimentally investigated the wake transition
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
regime of a square cylinder. The authors did not observe discontinuities on St-Re curve even though they identified type A and type B vortices in the spanwise region while reporting. The transition to mode A an B type instabilities, which had a spanwise length of 5.2D and 1.2D, took place at Ree160 and 200, respectively. Saha et al. [6] numerically studied the special evolution of vortices and transition to three-dimensionality in the wake of a square cylinder. They reported that three dimensionality takes place between 150 < Re < 175 and secondary vortices of mode A persist at 175 < Re < 240. B type vortices present in the wake region at about Ree250. Robichaux et al. [7] investigated the onset of threedimensionality in the wake of a square cylinder by applying Floquet stability analysis. They reported a long-wavelength (mode A) three-dimensional instabilities appears first at Ree161, followed by short-wavelength (mode B) instabilities at Ree190. In addition to these two modes, they observed intermediate-wavelength mode at Ree200. Their numerical analysis did not present discontinuity at the St-Re curve. In addition to the flow characteristics, two-dimensional forced convection heat transfer from a cylinder also was investigated by numerous researchers. One of these studies is performed by Sharma and Eswaran [8]. They investigated the flow structure and heat transfer characteristics of an isolated cylinder in the 2-D laminar flow regime for 1 Re ¼ 160 and Pr ¼ 0.7. They reported that the mean Nusselt number changed along the cylinder surface. The maximum Nusselt number is obtained at the front surface while it takes intermediate values at the top and bottom faces. The authors also proposed a heat transfer correlation applicable for 2-D flow regime. Recent study, on the flow and heat transfer across a circular cylinder has been performed by Golani and Dhimant [9] for Re ¼ 50e180 and Pr ¼ 0.7. They obtained variation of average Nu with Re. Karanth et al. [10] studied the effects of cylinder oscillation on the average Nusselt number. The heat transfer from the oscillating cylinder increased with the increasing velocity amplitude. The location of maximum local Nusselt number also oscillated between the upper and lower surface of the cylinder in the case of transverse oscillation. Two-dimensional laminar air flow (between 0.001 Re ¼ 170) and heat transfer past a circular isothermal cylinder was investigated by Shi et al. [11] with an emphasis on the heating effects on the flow characteristics. The numerical experiments with temperature dependent fluid properties resulted in depicting the effects of dynamic viscosity and density variations on the vortex shedding frequency. The effect of blockage on heat transfer from a square cylinder in a channel was studied by Turki et al. [12]. It was found that the flow was unstable when Richardson number crosses the critical value of 0.13. The heat transfer coefficient was influenced by thermally induced flow when Ri > 0.2. A more comprehensive investigation of the heating effects on the vortex structure in the wake was performed by Ren et al. [13], Van Steenhoven and Rindt [14], Badr [15] Van Steenhoven and Rindt [14] investigated 2-D behavior and 3-D transition behind a heated circular cylinder for Re ¼ 100. It was reported that the vortex street had undergone a negative deflection which caused by difference between the strengths of the upper and lower vortices at Ri<1. For Ri>1, an early 3-D transition with mushroom-type structure appear on top of the upper vortex row. The three-dimensional transition of a water flow around a heated circular cylinder was studied by Ren et al. [13]. Escaping mushroom-type structure in the far wake was observed for Re ¼ 85, Pr ¼ 7 and Ri ¼ 1 case. The origin of the type structure was the generation of streamwise vortices in the near wake.
Persillon and Braza [16] showed that the frequency discontinuities in Re-frequency curve are associated with a discontinuity in the local kinetic energy distribution in the near-wake region. Zhank et al. [17] observed four different physical instabilities namely vortex adhesion mode and three near-wake instabilities which are associated with three different spanwise wave-length. Although many experimental studies performed on the discontinuities of the St-Re curves, the foregoing numerical studies focused on investigation of the longwavelength, short-wavelength vortex structure in the spanwise direction of wake region, and variation of St number with the Re number. Although 3-D flow characteristics passed a cylinder at low Reynolds numbers have been extensively studied both experimentally and numerically, the numerical studies involving heat transfer from cylinders are based on 2-D simulations. The effects of the 3-D flow on the heat transfer characteristics have yet to be investigated. The aim of the present study is to determine the heat transfer characteristics from the cylinder in cross flow of air and explore the effects of the three dimensional flow. For this purpose, 3-D simulations were performed for Re ¼ 155e250 including transformation from 2-D flow and heat transfer to 3-D. The time histories of the mean Nusselt number evaluated over the cylinder surface area are computed. Instantaneous and mean deviation of the local Nusselt numbers on the cylinder surfaces both in spanwise direction and xy plane are also provided.
2. Governing equations and boundary conditions Dimensionless form of the unsteady Navier-Stokes equations for incompressible three-dimensional fluid flow are as follows:
vui ¼0 vxi
(1)
vui vu vp 1 v2 u i þ uj i ¼ þ vxi Re vxj vxj vt vxj
(2)
vq v uj q 1 v2 q þ ¼ RePr vxj vxj vt vxj
(3)
where ui, are velocity components, p is the pressure, and t is the dimensionless time. The dimensionless temperature q is defined as q ¼ ðΤ Τ∞ Þ=ðΤu Τ∞ Þ. The other dimensionless parameters governing the flow and heat transfer parameter defined as Re ¼ U∞D/n and Pr ¼ mcp/k where D is the side lenght of the square cylinder and U∞ is the free stream velocity at the inlet. The Prandtl number of air is assumed to be 0.7. The geometry and dimensions considered in this simulation is illustrated in Fig. 1(a) and (b). A fixed square-cylinder (D D 6D) is maintained at a constant temperature Tw, and it is placed in a free stream of velocity U and temperature T∞. The boundary conditions are as follows: For the inlet, top and bottom boundaries, u ¼ 1; v ¼ 0; q ¼ 0, vw vq For the top and bottom boundaries, v ¼ vu vy ¼ vy ¼ vy ¼ 0, v4 For the outlet, v4 vt þ uc vx ¼ 0, On the cylinders walls, u ¼ v ¼ w ¼ 0 (no slip). On the cylinder, qu ¼ 1 (isothermal wall).where 4 is all velocity components and temperature, uc is the velocity of the vortices leaving the outflow plane. The lift and drag coefficients of downstream cylinder are computed from
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
39
Fig. 1. The computational domain: (a) side (xez) view, (b) top (xey) view.
CL ¼
2Fy
rU2∞ D
;
CD ¼
2Fx
rU2∞ D
(4)
where Fx and Fy are the force components resolved in the directions x and y. The local heat transfer coefficient and the local Nusselt number are computed from
k
vT vn
w
¼ hs ðTw T∞ Þ;
Nus ¼
hs D k
(5)
where n is the direction normal to the cylinder surface; hs and Nus are the local heat transfer coefficient and local Nusselt number, respectively; and s is the circulation distance along the perimeter of the cylinder.
Fig. 2. The grid structure of computational domain in x-y plane.
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Table 1 The effects of the computational domain dimensions on the flow and heat transfer parameters for Re ¼ 250. (x1/y1 are distances with respect to cylinder center as shown in Fig. 1(b).). Case
x1/y1
A 25/10 B 20/7.5 C 15/6 Difference.between case A-B Differencebetween case B, C
CD,mean
CL,rms
Numean
St
1.583 1.597 1.633 0.9% 2.2%
0.77 0.745 0.814 3.3% 9%
6.010 6.021 6.062 0.2% 0.7%
0.142 0.142 0.144 0.0% 1.5%
Table 2 Effect of the grid refinement on the flow characteristic and heat transfer for Re ¼ 250. Case
Grid Nx Ny Nz
F 212 155 25 G 177 127 25 H 146 103 25 Difference between case F-G Difference between case G-H
dmin
Dt
CD,mean CL,rms
0.005 0.005 1.603 0.01 0.01 1.582 0.02 0.02 1.66 1.32% 4.93%
0.335 0.322 0.27 4.0% 16.1%
Numean St 5.644 5.611 5.872 0.6% 4.6%
0.152 0.153 0.158 0.65% 3.2%
cylinder. At x-z plane, uniform grids with Dzi ¼ 0.25D was used. The computations were carried out for a non-dimensional time step of Dt ¼ 0.01. In order to sufficiently observe the unsteady behavior of the drag and lift forces as well as the mean Nusselt number, simulation (dimensionless) time was extended up to 690. A fully implicit fractional step method has been employed to solve Eqs (1)e(3). All the terms in equations are advanced as follows:
u*i uni 1 v * * vpn 1 1 v v * ui uj þ uni uni ¼ þ þ u þ uni 2 vxj Dt vxi 2 Re vxj vxj i (6) v vpnþ1 1 vu*i ¼ vxi vxi Dt vxi
(7)
unþ1 u*i vfnþ1 i ¼ Dt vxi
(8)
pnþ1 ¼ pn þ f
(9)
where xi's are cartesian coordinates, ui's are corresponding velocity components, u*i ’s are the intermediate velocities.
3. Numerical details A non-uniform meshing in the x-y plane, but uniform in the x-z plane, was employed to the computation domain. The grid is clustered near the square cylinder as shown in Fig. 2. Near the walls of the cylinder, the width of the first cell in x-y plane was 0.01D. The distance between two consecutive grid lines was determined from a geometric series, with a stretching ratio of Dxi/Dxi-1 ¼ 0.9 in front and 1.05 behind the cylinder in x-direction, while in y-direction, Dyi/Dyi-1 ¼ 0.9 was employed to the bottom and 1.1 to the top of the
The convective terms including ðu*i u*j Þ are approximated as:
v * * v n * u u y u u vxj i j vxj i j
(10)
Staggered grid arrangement is employed to numerically solve Eq. (6)e(9). The first order spatial derivatives are discretized by applying a third-order upwind scheme while the second order derivatives of the viscous terms were discretized by employing the
Table 3 Verification of calculated St values. Re
Present study (3-D)
155 160 165 170 175 185 200 210 220 230 240 250
0.166 0.166 0.145 0.148 0.152 0.154 0.154 0.151 0.151 0.154 0.151 0.152
Robinchaux et al. [7]
Okajima [18] Exp.
Luo et al. [5] Exp.
Sohankar et al. [19]
Saha et al. [6]
Luo et al. [4] Exp.
0.156
0.164 0.165 0.159 0.159
0.165 0.142
0.157
0.143
0.159 0.159 0.159 0.161
0.16
0.166
0.143
0.159
0.157
0.145
0.163 0.16 0.165 0.164 0.164 0.164 0.163
Table 4 Verification of calculated mean drag coefficients. Re
Present study (3-D)
155 160 165 170 175 185 200 210 220 230 240 250
1.533 1.539 1.458 1.46 1.487 1.49 1.518 1.542 1.554 1.573 1.577 1.567
Robinchaux et al. [7]
Sohankar et al. [19]
Saha et al. [6]
Davis et al. [20] Exp.
1.53
1.7
1.58 1.6
1.4
1.59
1.43 1.72
1.67
1.78
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Fig. 3. For Re ¼ 185, vorticity (left column) and isotherms (right column) in the wake region.
Fig. 4. For Re ¼ 250, vorticity (left column) and isotherms (right column) in the wake region.
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central-difference formula. x, y and z component of the discretized momentum equations were solved separately using CrankNicholson time-stepping method. Poisson type equations are solved using Jacobi method with Chebyshev acceleration procedure. To determine the ideal computational domain which will reflect proper boundary locations, three computational domains, 15/6, 20/7.5, and 25/10 in x1 and y1 directions respectively, were tested for Re ¼ 250 by using the 2D-version of the current program. Table 1 depicts the test results for St, CDmean, CL,rms and Numean number. The Numean was obtained by taking the average of the time history of the Nusselt numbers on the up, down, front and rear surface of the cylinder. From the examination of the flow and heat transfer characteristics listed in Table 1, it was determined that Case B was sufficient enough to obtain meaningful and accurate results. A comparison of the effect of grid structure on the solution is illustrated in Table 2. Three different grid configurations were employed with various minimum grid spacing, dmin, near the solid boundaries. The number of grids in z-direction was kept constant at
25 for the tests. The time increment was 0.005, 0.01 and 0.02 for grids F, G and H respectively. Time step of Dt ¼ 0.01 was determined to be the optimum value to be used in the simulations. It is observed that the St Number is insensitive to the changes in grid configuration and minimum grid spacing at the cylinder wall. It was determined that Cases F and G provided sufficiently accurate results for the flow and heat transfer parameters, and in this investigation, Case G could be employed for the sake of cpu economy. The location of the cylinder and dimension of the computational domain presented at Fig. 1. The grid structure and grid refinement near the walls of the numerical model are also depicted in Fig. 2. The values of CD,mean and St numbers are obtained for various Re numbers and compared with the experimental and numerical data cited in the literature (Table 3 and Table 4). The experimental studies are indicated with “Exp.” labels. The numerical values for Refs. [4] and [6] are obtained by digitizing the corresponding St and CDmean curves. The computed St numbers are close to the experimental result of Luo et al. [4] (Table 3). There is discrepancy between CD,mean cited in the literature. This could be as a result of the
Fig. 5. Instantaneous temperature isotherms behind the cylinder for (a) Re ¼ 155, (b) Re ¼ 175, (c) Re ¼ 185, and (d) Re ¼ 250.
Fig. 6. Instantaneous Nusselt number variations on the surfaces of the cylinder on (a) font surface, (b) rear surface, (c) up surface, and (d) down surface for Re ¼ 185.
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
treatment of the cylinder edges, different experimental conditions at the experiments and the use of different numerical model at the simulations. However, the computed mean values are in the ranges of data given in the literature (Table 4).
4. Result and discussion 4.1. Vorticity structure and isotherms The instantaneous vorticity and isotherms in the wake region of the square cylinder are depicted in Figs. 3 and 4 for Re ¼ 185 and 250, respectively. Fig. 3(a) presents instantaneous vortex structure in x-y plane for z ¼ 3 and Re ¼ 185. Mode-A type vortices in the spanwise direction and positive vortex in the bottom side form while negative vortices are observed on top of the cylinder. Fig. 3(b) depicts the isotherms taken at the downstream of the cylinder at the same location and
43
instance as Fig. 3(a). This illustrates the variations of the isotherms which are compatible with the vortex structure in the wake region. Fig. 3(c) depicts the instantaneous vortex structure in spanwise direction at y ¼ 7.5. A-type vortices, positive vortex on the top and negative one on the bottom side, form just behind the cylinder. Fig. 3(d) corresponds to the isotherms at the downstream of cylinder at same instance. In the vicinity of the cylinder, the isotherms appear as straight lines; however, a close-view of this region reveals deflecting isotherms towards to downstream region. At the downstream region, the isotherms form columns of the closed circles and curved lines. The flow structure and isotherms for Re ¼ 250 are depicted in Fig. 4. At a higher Reynolds number, the flow structure at the downstream of cylinder turns to B-type vortices. Their variation patterns are similar in the x-y plane. Fig. 4(c) presents B-type smaller wave length (approximately 1D) vortices in the streamwise direction. The vorticity structure appears to be irregular, and positive/negative vortices form close to the
Fig. 7. Time history of Nusselt Numbers on left, up and right surface of the cylinder for Re ¼ 155.
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
cylinder. Further into the downstream region, the negative vortices become dominant. The isotherms correspond to the same instance depicted in Fig. 4(d). The isotherms appear as straight lines near the cylinder; but they depict a diverging trend at the downstream region at z ¼ 1.8, 3.2 ve 4.8 further downstream. At these locations, the temperature gradients are slightly smaller, and as a result, the heat transfer from the remaining part of the cylinder surface decreases.
4.2. Instantaneous isotherm surface and nusselt number variations Fig. 5 presents the isotherm surfaces at the downstream of the cylinder. High temperature isotherm surfaces remains close to cylinder, lower temperature isotherm surfaces begins close to cylinder at the front face however extends to further downstream region. The appearances of low temperature isotherm surface are similar to that of isovorticity curves in the downstream region (not shown here).
For Re ¼ 155, 2-D flow is characterized by smooth isotherm surfaces in spanwise direction (Fig. 5-a), however, for Re ¼ 175 and 185 where A-type vorticities exist, the wavy isotherm surfaces appear in spanwise direction (Fig. 5-b,c). For Re ¼ 250, the low temperature isotherm surfaces appears more mixed at further downstream region (Fig. 5-d). Fig. 6 presents the instantaneous variation of instantaneous Nusselt number along z-axis on the cylinder surfaces for Re ¼ 185. At the front surface, variation of the Nusselt number along z-axis is very smooth indicating that 3-D flow at the downstream region does not have obvious effect on the heat transfer from this surface. The oncoming fluid strikes the cylinder surface so that the heat transfer is larger. Heat transfer is relatively small at the cylinder front stagnation point. It takes large values up and down edges of the front surface where the boundary layer develops (Fig. 6-a). Boundary layer development continues along the up and down surfaces. Heat transfer decreases toward to rear surface (Fig. 6-b, c) where the flow separates from cylinder. The effects of the 3-D flow
Fig. 8. Time history of Nusselt Numbers on left, up and right surface of the cylinder for Re ¼ 185.
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
behind the cylinder are obvious on the heat transfer from rear surface so that alternating 3-D flow along z-axis leads to variation of heat transfer in spanwise direction (Fig. 6-d). The Nusselt number gains large values around the middle of its length.
4.3. Time history of Nusselt numbers Time history of Nusselt number are given in Figs. 7e9 on the surface of the cylinder for Re ¼ 155, 175 and 250 respectively. At Re ¼ 155 the flow is 2-D, the variation of Nusselt number on the cylinder becomes completely periodic (Fig. 7). Time history of the Nusselt number on the bottom surface of the cylinder was similar to that on the upper surface which is why it is not presented here. The mean Nusselt number time histories taken from 2D simulations for Re ¼ 175e250 are fully periodic for the upper, bottom, front and rear surface of the cylinder (not shown here). In Figs. 8 and 9, the effects of the 3-dimensional flow on the heat transfer
45
clearly visible. The variation of the Nusselt number is similar to the drag and lift coefficient variations with typical periods which one is at the vortex formation period, and another one with the period of 7e22 vortex formation period. The mean Nusselt numbers on the front and rear walls of the cylinder have a period half that of the vortex formation period while those on the upper and bottom surfaces have the same period of vortex formation. Although the history of the Nusselt number exhibits similar variation to that of drag coefficient, it seems that largest heat transfer rates correspond to maximum oscillations at the lift coefficients. By evaluating the time history of Nusselt number, the heat transfer process can be classified as high heat transfer level (HH) and low heat transfer level (LH). The instant of the high heat level varies with time of 10e17 vortex formation period differences for Re ¼ 185, however the variation becomes 7T-22T for Re ¼ 250. The Nusselt number takes its largest values at the front surface of the cylinder where the oncoming fluid hits the cylinder surface. Lower velocity flow at the
Fig. 9. Time history of Nusselt Numbers on left, up and right surface of the cylinder for Re ¼ 250.
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
wake region causes to smaller values of heat transfer from rear surface. Fig. 8 presents time history of Nusselt number on the cylinder surfaces for Re ¼ 185 at which A-mode vortex structure occurs at the spanwise direction. The largest deviations at the Nusselt numbers were observed at the rear surface of the cylinder as the largest value of 3.53 and the lowest value of 2.54. However, the differences between the largest and the lowest values are approximately 0.56 and 0.39 on the bottom and front surfaces respectively. On the cylinder surfaces, the maximum values of the Nusselt numbers do not occur at the same instances. There is between 0.13T and 2.1T time delay between the maximum values of the Nusselt numbers taken on front and upper surfaces. This time delay
becomes 1.1T-1.9T for upper and rear surfaces. The amplitude of oscillations of the Nusselt numbers also differs on the cylinder surfaces with the smallest values on the rear surface. Fig. 9 presents time history of the Nusselt number on the surface of the cylinder for Re ¼ 250 at which B-mode vortex structure take place at the spanwise direction. Wherein the period of the oscillations in the second period is more irregular than that for Re ¼ 185. For this case, the differences between the maximum Nusselts numbers on the front, upper and bottom surfaces very close to those of for Re ¼ 185 (0.385, 0.416 and 1.58 on front, upper and rear surfaces respectively). However, at the rear surface, these differences are higher. Similar to those for Re ¼ 185, the mean Nusselt numbers of the upper and bottom faces have the same period of the
Fig. 10. The variation of the mean Nusselt Number at the left (a) up, and right (b) surface of the cylinder.
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
47
simulations are completely periodic (not presented here).
4.4. Variation of mean Nusselt number on cylinder surfaces with Reynolds numbers
Fig. 11. The variation of the mean local Nusselt number on the surface of the cylinder at z ¼ 3 (from average of 30 files).
vortex shedding, and the periods of the front and rear faces are half of the vortex shedding periods. The time history of the mean Nusselt numbers on cylinder surfaces obtained from 2-D
Fig. 10 compares the variation of mean Nusselt numbers on the cylinder walls for 2-D and 3-D simulations. On the front wall of the cylinder, the numerical predictions of 2-D and 3-D simulations are very close except at Re ¼ 220 where the flow conditions near the set of B-type instability. In this study, the mean Nusselt numbers obtained on the front face are slightly larger than that of Turki's (2003) result (Fig. 10(a)). However, 2-D simulations predict larger mean Nusselt numbers with a linear variation at the rear wall of the cylinder (Fig. 10(b)). At the rear wall of the cylinder, 3-D simulation yield a drop in the Nu values at Re ¼ 165 where the flow changes from 2-D to 3-D with A-type instability; on the other hand, it increases in the interval 165 Re ¼ 240. At the rear surface, the mean Nu values are larger than those of Turki's [12] at the 2-D flow region (Re < 165) and close to their prediction at 3-D flow region. At the upper surface, the mean Nusselt number slightly changes with Re number and 3-D simulation results are lower than those of the 2-D simulations and Turki's [12].
Fig. 12. The variation of the mean local Nusselt number on the cylinder surfaces along the spanwise direction at the midpoint of the surfaces (Re ¼ 185, from average of 30 files).
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
4.5. Variation of local Nusselt number
9.64 9.636
front surface
9.632 9.628 9.624 2.48 2.476 2.472 bottom surface
2.468 2.464 4.4 4 3.6 3.2 2.8 2.4 2.48
Nusselt Number
Nusselt Number
Nusselt Number
Nusselt Number
In Fig. 11, the variation of mean values of local Nusselt number on the cylinder walls are presented. The variation of mean values are computed by taking the average of 30 data files obtained at different times at the middle point of the cylinder z-axis (z ¼ 3). The largest Nusselt number values were obtained at the front wall. At the front stagnation point, these values are relatively small and take largest values at the end points of the front face. At the upper and bottom surfaces, the variation of mean Nusselt numbers are similar and takes the largest values at the edges which are close to rear surface, and the mean Nusselt number takes small values at the rear face. Fig. 11 also compares the local Nusselt number variations for Re ¼ 160, 185 and 250. For Re ¼ 160, variation local Nusselt number on the front surface is slightly underestimated than that provided by Sahu et al. [21] while the local Nusselt distributions on the other surfaces are observed to be identical. This deviation can be attributed to the choice of Prandtl numbers as Sahu's study [21] assumes Pr ¼ 1, this study uses Pr ¼ 0.7. For Re ¼ 250, the mean Nu number are larger on the front face and the effects of the Re number is much less on the other surfaces. In Figs. 12 and 13 illustrate the variations of the local Nusselt number on the cylinder walls in the streamwise direction for
Re ¼ 185 and 250 chosen representative of typical A and B type instability. The mean values are obtained at the mid-point of the cylinder at x-,y-axis. The mean local Nusselt number data presented are obtained by taking average of 30 data-files taken at different times on the walls of the cylinder. In Fig. 12, for Re ¼ 185, the variation of mean Nusselt number along z axis presents a deviation similar to sine curve. The maximum value of the Nusselt number is obtained at z ¼ 4.6 on the front, bottom and upper walls while it takes minimum value at z ¼ 2.65. However, the heat transfer rate becomes minimum at z ¼ 0.96 and maximum at z ¼ 2.4 on the rear face. There is 3.6D distance between the top points of the sinusoidal variation of the Nusselt number on front, upper and bottom faces while the same distances exist between the minimum points of Nusselt number variation on the rear surface. It is noticed that this distance is close to A-mode vortex formation wave length (z3D) along z-axis. The variation of the mean Nusselt number for Re ¼ 250, where B-mode exist in spanwise direction, is given at Fig. 13. Here, the variations of Nusselt numbers were irregular. The variation on upper and bottom surfaces are somewhat similar. The maximum heat transfer rate on the front surface take place at z ¼ 2.8 while it is at z ¼ 3.75 on the other faces. Deviation of the Nusselt number from its mean values in z-axis are very low on the front, upper and
rear surface u r
f
2.47 2.46
b
upper surface
2.45 2.44
0
2
4
6
z Fig. 13. The variation of the mean local Nusselt number on the cylinder surfaces along the spanwise direction at the midpoint of the surfaces (Re ¼ 250, from average of 30 files).
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
bottom surfaces; however, on the rear surface significant deviations occur. Fig. 14 presents instantaneous isotherms behind the cylinder corresponding to vortex structure given at Fig. 3(c) and instantaneous mean Nusselt number variation along z-axis. The isotherms appears dense at z ¼ 2.9 where maximum heat transfer rate take place while minimum heat transfer rates appears at z ¼ 1.2 and 4.8 with the coarse isotherm curves near the cylinder. For Re ¼ 250, instantaneous isotherms corresponds to Fig. 4(c) and variation of
Fig. 14. Close view of the isotherms at the rear face of the cylinder at the same instance with vortex structure shown in Fig. 3c (a) and corresponding local Nusselt number variation (b) (Re ¼ 185).
49
instantaneous local Nusselt number along z-axis is given in Fig. 15. At z ¼ 0.8, 2.4 and 3.8, the isotherms at the vicinity of the cylinder are more condensed, and the peaks at the mean Nusselt numbers distribution appears at these locations. The variation of the mean Nusselt number for Re ¼ 250, where B-mode exist in spanwise direction, is given at Fig. 13. Here, the variations of Nusselt numbers were irregular. 5. Conclusions In this study, three-dimensional unsteady flow and heat transfer from an isothermal square cylinder in crossflow of air is numerically investigated. The governing continuity, momentum and energy equations are solved numerically via implicit fractional step method where a third-order upwind scheme in first order derivatives is adapted. The heat transfer characteristics from a square cylinder in cross flow of air and the effects of 3D flow was assessed for Re ¼ 185 and 250. The study yield the following conclusion: In this study also A-mode and B-mode type flow regimes were identified by instantaneous vorticity lines at the wake of the cylinder and time history of the mean Nusselt numbers on the cylinder surfaces obtained. The isotherms and vorticity curves presents similarity in x-y plane while their view along z-axis differ. Time history of Nusselt number also differs on the cylinder faces with time delays between their maximum values on the cylinder faces. The time delay is more pronounced with 0.13Te2.1T at Re ¼ 185 where A-type flow mode exist in the wake. The variation of the mean Nusselt number along z-axis also differs depending on the flow regimes in spanwise direction. At Re ¼ 185 (A-mode flow), the distribution is approximately sinusoidal while it is less regular at Re ¼ 250 (B-mode). At both flow regimes, the amplitudes of the oscillations of Nusselt number are very small on front, up and down faces while they have relatively large amplitudes on the rear face. The heat transfer rates are larger where the temperature isosurfaces are more condensed which appear at the back face of the cylinder. At Re ¼ 250 (B-mode), the difference between these larger heat transfer locations vary as 1.4D-1.7D. Acknowledgment The author gratefully acknowledges the financial support by the Research Fund of Eskis¸ehir Osmangazi University, Project #201415039. References
Fig. 15. Close view of the isotherms at the rear surface of the cylinder at the same instance with vortex structure shown in Fig. 4c (a), and corresponding local Nusselt number variation (b) (Re ¼ 250).
[1] Williamson CHK. Defining a universal and continuous Strouhal-Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys Fluids 1988;31:2742e4. [2] Williamson CHK. The existence of two stages in the transition to threedimensionality. Phys Fluids 1988;31:3165e8. [3] Williamson CHK. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. Int J Fluid Mech 1989;206: 579e627. [4] Luo SC, Tong XH, Khoo BC. Transition phenomena in the wake of a square cylinder. J Fluid Struct 2007;23:227e48. [5] Luo SC, Chew YT, Ng YT. Characteristic of square cylinder wake transition flows. Phys Fluids 2003;15:2549e59. [6] Saha AK, Biswas G, Muralidhar K. Three-dimensional study of flow past a square cylinder at low Reynolds number. Int J Heat Fluid Flows 2003;24: 54e66. [7] Robichaux J, Balachandar S, Vanka SP. Three-dimensional floquet instability of the wake of square cylinder. Phys Fluids 1999;11:560e78. [8] Sharma A, Eswaran V. Heat and Fluid flow across a square cylinder in the twodimensional laminar flow regime. Numer Heat Transf Part A 2004;45:247e69. [9] Golani R, Dhiman AK. Fluid flow and heat transfer across a circular cylinder in the unsteady flow regime. Int J Eng Sci 2014;3:8e19. [10] Karanth D, Rankin GW, Sridhar K. A finite difference calculation of forced convective heat transfer from an oscillating cylinder. Int J Heat Mass Transf 1994;37:1619e30.
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[11] Shi J-M, Gerlach D, Bauer M. Heating effect on steady and unsteady horizontal laminar flow of air past a circular cylinder. Phys Fluids 2004;16:4331e45. [12] Turki S, Abbassi H, Nasrallah SB. Two-dimensional laminar fluid flow and heat transfer in a channel with a build-in heated square cylinder. Int J Therm Scieces 2003;42:1105e13. [13] Ren M, Rindt CCM, Van Steenhoven AA. Three-dimensional transition of a water flow around a heated cylinder at Re¼85 and Ri ¼ 1.0. J Fluid Mech 2006;566:195e224. [14] Van Steenhoven AA, Rindt CCM. Flow transition behind a heated cylinder. Int J Heat Fluid Flow 2003;24:323e33. [15] Badr H. Laminar combined convection from a horizontal cylinder-parallel and contra flow regimes. Intl J Heat Mass Transf 1984;27:15e27. [16] Persillon H, Braza M. Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional Navier-Stokes simulation.
J Fluid Mech 1988;365:23e88. €nig M, Eckelmann H. On the transition of the [17] Zhang HQ, Fey U, Noack BR, Ko cylinder wake. Phys Fluids 1995;7:779e94. [18] Okajima A. Strouhal numbers of rectangular cylinders. J Fluid Mech 1982;123: 379e89. [19] Sohankar A, Norberg C, Davidson L. Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers. Phys Fluids 1999;11:288e306. [20] Davis RW, Moore EF, Purtell LP. A numerical-experimental study of confined flow around rectangular cylinders. Phys Fluids 1984;27:46e59. [21] Sahu Akhilesh K, Chhabra RP, Eswaran V. Effects of Reynolds and Prandtl numbers on heat transfer from a square cylinder in the unsteady flow regime. Int J Heat Mass Transf 2009;52:839e50.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Three dimensional heat transfer from a square cylinder at low Reynolds numbers Necati Mahir Eskis¸ehir Osmangazi University, School of Engineering and Architecture, Mechanical Engineering Department, 26480 Batı Mes¸elik, Eskis¸ehir, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 January 2017 Received in revised form 26 April 2017 Accepted 27 April 2017 Available online 1 June 2017
Three-dimensional unsteady flow and heat transfer from an isothermal square cylinder subjected to crossflow of air is numerically investigated. The governing continuity, momentum and energy equations are solved using implicit fractional step method. The first order spatial derivatives are discretized using a third-order upwind scheme while the second order derivatives from the viscous terms are discretized by central-difference formula. The momentum equations are solved separately using Crank-Nicholson timestepping method. The Poisson type equations are solved using Jacobi method with Chebyshev acceleration procedure. The heat transfer characteristics of a square cylinder subjected cross flow of air and the threedimensionality of the flow and its effects are assessed. Three dimensional instantaneous isotemperature surfaces and in the wake region and local Nusselt number variations on the cylinder surfaces along z-axis were provided. The time histories of the Nusselt numbers at the cylinder surfaces are obtained. Also instantaneous and mean deviation of the local Nusselt numbers on the cylinder surfaces both in spanwise direction and x-y plane are provided for Re ¼ 185 and 250 which are representative of typical A-mode and B-mode vortex structures in spanwise direction respectively. © 2017 Elsevier Masson SAS. All rights reserved.
Keywords: 3-D heat transfer Square cylinder Forced convection
1. Introduction The fluid flow and heat transfer around a bluff body has received increasing interest in many engineering fields due to its applications such as heat exchangers, cooling systems, cooling of electronic systems, pipelines, flow around nuclear rods, heat lost from buildings, etc. Therefore, flow around the unconfined circular and/ or square cylinders has been experimentally and numerically studied for the last decades. It has been long established that the fluid flow around a cylinder, at small Reynolds numbers (Re < 165), is laminar and twodimensional. At relatively large Reynolds numbers (Re > 165), the flow becomes three-dimensional having well-known discontinuities at the St-Re curve. This discontinuity was reported by Williamson [1e3], Luo et al. [4] and others. The transition phenomena in the wake of a circular cylinder were studied in detail by Luo et al. [4]. Mode A and mode B type instabilities in the wake of a square cylinder were studied and determined by identifying the discontinuities in the St-Re curves.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijthermalsci.2017.04.031 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.
The authors found that Reynolds number had mean values of 160 and 204 for the onset of the mode A and B type instabilities, respectively. The experimental study of Luo et al. [4] also showed that the aspect ratio of the cylinder and the end plates which used to promote a three-dimensional flow have a strong influence on the wake of a cylinder. They obtained the St-Re discontinuity by resolving the flow for sufficiently small increments of the Reynolds number. It was shown that when the cylinder aspect ratio was less than 22.2, the magnitude of the Strouhal number increases with the cylinder aspect ratio and remains almost constant for larger aspect ratios. Large scale vortex shedding in the wake region and the existence of discontinuities in the St-Re relationships was studied by Williamson [1e3]. It was found that both of these wake characteristics are directly related to each other. The St discontinuities are caused by a transition from one oblique shedding mode to another oblique mode. Vortex dislocation is observed due to vortices in neighboring cells which move out of phase with each other. However, by manipulating the spanwise end conditions of the cylinder, it becomes possible to induce parallel vortex shedding which results in continuous St-Re curve. Luo et al. [5] experimentally investigated the wake transition
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
regime of a square cylinder. The authors did not observe discontinuities on St-Re curve even though they identified type A and type B vortices in the spanwise region while reporting. The transition to mode A an B type instabilities, which had a spanwise length of 5.2D and 1.2D, took place at Ree160 and 200, respectively. Saha et al. [6] numerically studied the special evolution of vortices and transition to three-dimensionality in the wake of a square cylinder. They reported that three dimensionality takes place between 150 < Re < 175 and secondary vortices of mode A persist at 175 < Re < 240. B type vortices present in the wake region at about Ree250. Robichaux et al. [7] investigated the onset of threedimensionality in the wake of a square cylinder by applying Floquet stability analysis. They reported a long-wavelength (mode A) three-dimensional instabilities appears first at Ree161, followed by short-wavelength (mode B) instabilities at Ree190. In addition to these two modes, they observed intermediate-wavelength mode at Ree200. Their numerical analysis did not present discontinuity at the St-Re curve. In addition to the flow characteristics, two-dimensional forced convection heat transfer from a cylinder also was investigated by numerous researchers. One of these studies is performed by Sharma and Eswaran [8]. They investigated the flow structure and heat transfer characteristics of an isolated cylinder in the 2-D laminar flow regime for 1 Re ¼ 160 and Pr ¼ 0.7. They reported that the mean Nusselt number changed along the cylinder surface. The maximum Nusselt number is obtained at the front surface while it takes intermediate values at the top and bottom faces. The authors also proposed a heat transfer correlation applicable for 2-D flow regime. Recent study, on the flow and heat transfer across a circular cylinder has been performed by Golani and Dhimant [9] for Re ¼ 50e180 and Pr ¼ 0.7. They obtained variation of average Nu with Re. Karanth et al. [10] studied the effects of cylinder oscillation on the average Nusselt number. The heat transfer from the oscillating cylinder increased with the increasing velocity amplitude. The location of maximum local Nusselt number also oscillated between the upper and lower surface of the cylinder in the case of transverse oscillation. Two-dimensional laminar air flow (between 0.001 Re ¼ 170) and heat transfer past a circular isothermal cylinder was investigated by Shi et al. [11] with an emphasis on the heating effects on the flow characteristics. The numerical experiments with temperature dependent fluid properties resulted in depicting the effects of dynamic viscosity and density variations on the vortex shedding frequency. The effect of blockage on heat transfer from a square cylinder in a channel was studied by Turki et al. [12]. It was found that the flow was unstable when Richardson number crosses the critical value of 0.13. The heat transfer coefficient was influenced by thermally induced flow when Ri > 0.2. A more comprehensive investigation of the heating effects on the vortex structure in the wake was performed by Ren et al. [13], Van Steenhoven and Rindt [14], Badr [15] Van Steenhoven and Rindt [14] investigated 2-D behavior and 3-D transition behind a heated circular cylinder for Re ¼ 100. It was reported that the vortex street had undergone a negative deflection which caused by difference between the strengths of the upper and lower vortices at Ri<1. For Ri>1, an early 3-D transition with mushroom-type structure appear on top of the upper vortex row. The three-dimensional transition of a water flow around a heated circular cylinder was studied by Ren et al. [13]. Escaping mushroom-type structure in the far wake was observed for Re ¼ 85, Pr ¼ 7 and Ri ¼ 1 case. The origin of the type structure was the generation of streamwise vortices in the near wake.
Persillon and Braza [16] showed that the frequency discontinuities in Re-frequency curve are associated with a discontinuity in the local kinetic energy distribution in the near-wake region. Zhank et al. [17] observed four different physical instabilities namely vortex adhesion mode and three near-wake instabilities which are associated with three different spanwise wave-length. Although many experimental studies performed on the discontinuities of the St-Re curves, the foregoing numerical studies focused on investigation of the longwavelength, short-wavelength vortex structure in the spanwise direction of wake region, and variation of St number with the Re number. Although 3-D flow characteristics passed a cylinder at low Reynolds numbers have been extensively studied both experimentally and numerically, the numerical studies involving heat transfer from cylinders are based on 2-D simulations. The effects of the 3-D flow on the heat transfer characteristics have yet to be investigated. The aim of the present study is to determine the heat transfer characteristics from the cylinder in cross flow of air and explore the effects of the three dimensional flow. For this purpose, 3-D simulations were performed for Re ¼ 155e250 including transformation from 2-D flow and heat transfer to 3-D. The time histories of the mean Nusselt number evaluated over the cylinder surface area are computed. Instantaneous and mean deviation of the local Nusselt numbers on the cylinder surfaces both in spanwise direction and xy plane are also provided.
2. Governing equations and boundary conditions Dimensionless form of the unsteady Navier-Stokes equations for incompressible three-dimensional fluid flow are as follows:
vui ¼0 vxi
(1)
vui vu vp 1 v2 u i þ uj i ¼ þ vxi Re vxj vxj vt vxj
(2)
vq v uj q 1 v2 q þ ¼ RePr vxj vxj vt vxj
(3)
where ui, are velocity components, p is the pressure, and t is the dimensionless time. The dimensionless temperature q is defined as q ¼ ðΤ Τ∞ Þ=ðΤu Τ∞ Þ. The other dimensionless parameters governing the flow and heat transfer parameter defined as Re ¼ U∞D/n and Pr ¼ mcp/k where D is the side lenght of the square cylinder and U∞ is the free stream velocity at the inlet. The Prandtl number of air is assumed to be 0.7. The geometry and dimensions considered in this simulation is illustrated in Fig. 1(a) and (b). A fixed square-cylinder (D D 6D) is maintained at a constant temperature Tw, and it is placed in a free stream of velocity U and temperature T∞. The boundary conditions are as follows: For the inlet, top and bottom boundaries, u ¼ 1; v ¼ 0; q ¼ 0, vw vq For the top and bottom boundaries, v ¼ vu vy ¼ vy ¼ vy ¼ 0, v4 For the outlet, v4 vt þ uc vx ¼ 0, On the cylinders walls, u ¼ v ¼ w ¼ 0 (no slip). On the cylinder, qu ¼ 1 (isothermal wall).where 4 is all velocity components and temperature, uc is the velocity of the vortices leaving the outflow plane. The lift and drag coefficients of downstream cylinder are computed from
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
39
Fig. 1. The computational domain: (a) side (xez) view, (b) top (xey) view.
CL ¼
2Fy
rU2∞ D
;
CD ¼
2Fx
rU2∞ D
(4)
where Fx and Fy are the force components resolved in the directions x and y. The local heat transfer coefficient and the local Nusselt number are computed from
k
vT vn
w
¼ hs ðTw T∞ Þ;
Nus ¼
hs D k
(5)
where n is the direction normal to the cylinder surface; hs and Nus are the local heat transfer coefficient and local Nusselt number, respectively; and s is the circulation distance along the perimeter of the cylinder.
Fig. 2. The grid structure of computational domain in x-y plane.
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
Table 1 The effects of the computational domain dimensions on the flow and heat transfer parameters for Re ¼ 250. (x1/y1 are distances with respect to cylinder center as shown in Fig. 1(b).). Case
x1/y1
A 25/10 B 20/7.5 C 15/6 Difference.between case A-B Differencebetween case B, C
CD,mean
CL,rms
Numean
St
1.583 1.597 1.633 0.9% 2.2%
0.77 0.745 0.814 3.3% 9%
6.010 6.021 6.062 0.2% 0.7%
0.142 0.142 0.144 0.0% 1.5%
Table 2 Effect of the grid refinement on the flow characteristic and heat transfer for Re ¼ 250. Case
Grid Nx Ny Nz
F 212 155 25 G 177 127 25 H 146 103 25 Difference between case F-G Difference between case G-H
dmin
Dt
CD,mean CL,rms
0.005 0.005 1.603 0.01 0.01 1.582 0.02 0.02 1.66 1.32% 4.93%
0.335 0.322 0.27 4.0% 16.1%
Numean St 5.644 5.611 5.872 0.6% 4.6%
0.152 0.153 0.158 0.65% 3.2%
cylinder. At x-z plane, uniform grids with Dzi ¼ 0.25D was used. The computations were carried out for a non-dimensional time step of Dt ¼ 0.01. In order to sufficiently observe the unsteady behavior of the drag and lift forces as well as the mean Nusselt number, simulation (dimensionless) time was extended up to 690. A fully implicit fractional step method has been employed to solve Eqs (1)e(3). All the terms in equations are advanced as follows:
u*i uni 1 v * * vpn 1 1 v v * ui uj þ uni uni ¼ þ þ u þ uni 2 vxj Dt vxi 2 Re vxj vxj i (6) v vpnþ1 1 vu*i ¼ vxi vxi Dt vxi
(7)
unþ1 u*i vfnþ1 i ¼ Dt vxi
(8)
pnþ1 ¼ pn þ f
(9)
where xi's are cartesian coordinates, ui's are corresponding velocity components, u*i ’s are the intermediate velocities.
3. Numerical details A non-uniform meshing in the x-y plane, but uniform in the x-z plane, was employed to the computation domain. The grid is clustered near the square cylinder as shown in Fig. 2. Near the walls of the cylinder, the width of the first cell in x-y plane was 0.01D. The distance between two consecutive grid lines was determined from a geometric series, with a stretching ratio of Dxi/Dxi-1 ¼ 0.9 in front and 1.05 behind the cylinder in x-direction, while in y-direction, Dyi/Dyi-1 ¼ 0.9 was employed to the bottom and 1.1 to the top of the
The convective terms including ðu*i u*j Þ are approximated as:
v * * v n * u u y u u vxj i j vxj i j
(10)
Staggered grid arrangement is employed to numerically solve Eq. (6)e(9). The first order spatial derivatives are discretized by applying a third-order upwind scheme while the second order derivatives of the viscous terms were discretized by employing the
Table 3 Verification of calculated St values. Re
Present study (3-D)
155 160 165 170 175 185 200 210 220 230 240 250
0.166 0.166 0.145 0.148 0.152 0.154 0.154 0.151 0.151 0.154 0.151 0.152
Robinchaux et al. [7]
Okajima [18] Exp.
Luo et al. [5] Exp.
Sohankar et al. [19]
Saha et al. [6]
Luo et al. [4] Exp.
0.156
0.164 0.165 0.159 0.159
0.165 0.142
0.157
0.143
0.159 0.159 0.159 0.161
0.16
0.166
0.143
0.159
0.157
0.145
0.163 0.16 0.165 0.164 0.164 0.164 0.163
Table 4 Verification of calculated mean drag coefficients. Re
Present study (3-D)
155 160 165 170 175 185 200 210 220 230 240 250
1.533 1.539 1.458 1.46 1.487 1.49 1.518 1.542 1.554 1.573 1.577 1.567
Robinchaux et al. [7]
Sohankar et al. [19]
Saha et al. [6]
Davis et al. [20] Exp.
1.53
1.7
1.58 1.6
1.4
1.59
1.43 1.72
1.67
1.78
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
Fig. 3. For Re ¼ 185, vorticity (left column) and isotherms (right column) in the wake region.
Fig. 4. For Re ¼ 250, vorticity (left column) and isotherms (right column) in the wake region.
41
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
central-difference formula. x, y and z component of the discretized momentum equations were solved separately using CrankNicholson time-stepping method. Poisson type equations are solved using Jacobi method with Chebyshev acceleration procedure. To determine the ideal computational domain which will reflect proper boundary locations, three computational domains, 15/6, 20/7.5, and 25/10 in x1 and y1 directions respectively, were tested for Re ¼ 250 by using the 2D-version of the current program. Table 1 depicts the test results for St, CDmean, CL,rms and Numean number. The Numean was obtained by taking the average of the time history of the Nusselt numbers on the up, down, front and rear surface of the cylinder. From the examination of the flow and heat transfer characteristics listed in Table 1, it was determined that Case B was sufficient enough to obtain meaningful and accurate results. A comparison of the effect of grid structure on the solution is illustrated in Table 2. Three different grid configurations were employed with various minimum grid spacing, dmin, near the solid boundaries. The number of grids in z-direction was kept constant at
25 for the tests. The time increment was 0.005, 0.01 and 0.02 for grids F, G and H respectively. Time step of Dt ¼ 0.01 was determined to be the optimum value to be used in the simulations. It is observed that the St Number is insensitive to the changes in grid configuration and minimum grid spacing at the cylinder wall. It was determined that Cases F and G provided sufficiently accurate results for the flow and heat transfer parameters, and in this investigation, Case G could be employed for the sake of cpu economy. The location of the cylinder and dimension of the computational domain presented at Fig. 1. The grid structure and grid refinement near the walls of the numerical model are also depicted in Fig. 2. The values of CD,mean and St numbers are obtained for various Re numbers and compared with the experimental and numerical data cited in the literature (Table 3 and Table 4). The experimental studies are indicated with “Exp.” labels. The numerical values for Refs. [4] and [6] are obtained by digitizing the corresponding St and CDmean curves. The computed St numbers are close to the experimental result of Luo et al. [4] (Table 3). There is discrepancy between CD,mean cited in the literature. This could be as a result of the
Fig. 5. Instantaneous temperature isotherms behind the cylinder for (a) Re ¼ 155, (b) Re ¼ 175, (c) Re ¼ 185, and (d) Re ¼ 250.
Fig. 6. Instantaneous Nusselt number variations on the surfaces of the cylinder on (a) font surface, (b) rear surface, (c) up surface, and (d) down surface for Re ¼ 185.
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
treatment of the cylinder edges, different experimental conditions at the experiments and the use of different numerical model at the simulations. However, the computed mean values are in the ranges of data given in the literature (Table 4).
4. Result and discussion 4.1. Vorticity structure and isotherms The instantaneous vorticity and isotherms in the wake region of the square cylinder are depicted in Figs. 3 and 4 for Re ¼ 185 and 250, respectively. Fig. 3(a) presents instantaneous vortex structure in x-y plane for z ¼ 3 and Re ¼ 185. Mode-A type vortices in the spanwise direction and positive vortex in the bottom side form while negative vortices are observed on top of the cylinder. Fig. 3(b) depicts the isotherms taken at the downstream of the cylinder at the same location and
43
instance as Fig. 3(a). This illustrates the variations of the isotherms which are compatible with the vortex structure in the wake region. Fig. 3(c) depicts the instantaneous vortex structure in spanwise direction at y ¼ 7.5. A-type vortices, positive vortex on the top and negative one on the bottom side, form just behind the cylinder. Fig. 3(d) corresponds to the isotherms at the downstream of cylinder at same instance. In the vicinity of the cylinder, the isotherms appear as straight lines; however, a close-view of this region reveals deflecting isotherms towards to downstream region. At the downstream region, the isotherms form columns of the closed circles and curved lines. The flow structure and isotherms for Re ¼ 250 are depicted in Fig. 4. At a higher Reynolds number, the flow structure at the downstream of cylinder turns to B-type vortices. Their variation patterns are similar in the x-y plane. Fig. 4(c) presents B-type smaller wave length (approximately 1D) vortices in the streamwise direction. The vorticity structure appears to be irregular, and positive/negative vortices form close to the
Fig. 7. Time history of Nusselt Numbers on left, up and right surface of the cylinder for Re ¼ 155.
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
cylinder. Further into the downstream region, the negative vortices become dominant. The isotherms correspond to the same instance depicted in Fig. 4(d). The isotherms appear as straight lines near the cylinder; but they depict a diverging trend at the downstream region at z ¼ 1.8, 3.2 ve 4.8 further downstream. At these locations, the temperature gradients are slightly smaller, and as a result, the heat transfer from the remaining part of the cylinder surface decreases.
4.2. Instantaneous isotherm surface and nusselt number variations Fig. 5 presents the isotherm surfaces at the downstream of the cylinder. High temperature isotherm surfaces remains close to cylinder, lower temperature isotherm surfaces begins close to cylinder at the front face however extends to further downstream region. The appearances of low temperature isotherm surface are similar to that of isovorticity curves in the downstream region (not shown here).
For Re ¼ 155, 2-D flow is characterized by smooth isotherm surfaces in spanwise direction (Fig. 5-a), however, for Re ¼ 175 and 185 where A-type vorticities exist, the wavy isotherm surfaces appear in spanwise direction (Fig. 5-b,c). For Re ¼ 250, the low temperature isotherm surfaces appears more mixed at further downstream region (Fig. 5-d). Fig. 6 presents the instantaneous variation of instantaneous Nusselt number along z-axis on the cylinder surfaces for Re ¼ 185. At the front surface, variation of the Nusselt number along z-axis is very smooth indicating that 3-D flow at the downstream region does not have obvious effect on the heat transfer from this surface. The oncoming fluid strikes the cylinder surface so that the heat transfer is larger. Heat transfer is relatively small at the cylinder front stagnation point. It takes large values up and down edges of the front surface where the boundary layer develops (Fig. 6-a). Boundary layer development continues along the up and down surfaces. Heat transfer decreases toward to rear surface (Fig. 6-b, c) where the flow separates from cylinder. The effects of the 3-D flow
Fig. 8. Time history of Nusselt Numbers on left, up and right surface of the cylinder for Re ¼ 185.
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
behind the cylinder are obvious on the heat transfer from rear surface so that alternating 3-D flow along z-axis leads to variation of heat transfer in spanwise direction (Fig. 6-d). The Nusselt number gains large values around the middle of its length.
4.3. Time history of Nusselt numbers Time history of Nusselt number are given in Figs. 7e9 on the surface of the cylinder for Re ¼ 155, 175 and 250 respectively. At Re ¼ 155 the flow is 2-D, the variation of Nusselt number on the cylinder becomes completely periodic (Fig. 7). Time history of the Nusselt number on the bottom surface of the cylinder was similar to that on the upper surface which is why it is not presented here. The mean Nusselt number time histories taken from 2D simulations for Re ¼ 175e250 are fully periodic for the upper, bottom, front and rear surface of the cylinder (not shown here). In Figs. 8 and 9, the effects of the 3-dimensional flow on the heat transfer
45
clearly visible. The variation of the Nusselt number is similar to the drag and lift coefficient variations with typical periods which one is at the vortex formation period, and another one with the period of 7e22 vortex formation period. The mean Nusselt numbers on the front and rear walls of the cylinder have a period half that of the vortex formation period while those on the upper and bottom surfaces have the same period of vortex formation. Although the history of the Nusselt number exhibits similar variation to that of drag coefficient, it seems that largest heat transfer rates correspond to maximum oscillations at the lift coefficients. By evaluating the time history of Nusselt number, the heat transfer process can be classified as high heat transfer level (HH) and low heat transfer level (LH). The instant of the high heat level varies with time of 10e17 vortex formation period differences for Re ¼ 185, however the variation becomes 7T-22T for Re ¼ 250. The Nusselt number takes its largest values at the front surface of the cylinder where the oncoming fluid hits the cylinder surface. Lower velocity flow at the
Fig. 9. Time history of Nusselt Numbers on left, up and right surface of the cylinder for Re ¼ 250.
46
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wake region causes to smaller values of heat transfer from rear surface. Fig. 8 presents time history of Nusselt number on the cylinder surfaces for Re ¼ 185 at which A-mode vortex structure occurs at the spanwise direction. The largest deviations at the Nusselt numbers were observed at the rear surface of the cylinder as the largest value of 3.53 and the lowest value of 2.54. However, the differences between the largest and the lowest values are approximately 0.56 and 0.39 on the bottom and front surfaces respectively. On the cylinder surfaces, the maximum values of the Nusselt numbers do not occur at the same instances. There is between 0.13T and 2.1T time delay between the maximum values of the Nusselt numbers taken on front and upper surfaces. This time delay
becomes 1.1T-1.9T for upper and rear surfaces. The amplitude of oscillations of the Nusselt numbers also differs on the cylinder surfaces with the smallest values on the rear surface. Fig. 9 presents time history of the Nusselt number on the surface of the cylinder for Re ¼ 250 at which B-mode vortex structure take place at the spanwise direction. Wherein the period of the oscillations in the second period is more irregular than that for Re ¼ 185. For this case, the differences between the maximum Nusselts numbers on the front, upper and bottom surfaces very close to those of for Re ¼ 185 (0.385, 0.416 and 1.58 on front, upper and rear surfaces respectively). However, at the rear surface, these differences are higher. Similar to those for Re ¼ 185, the mean Nusselt numbers of the upper and bottom faces have the same period of the
Fig. 10. The variation of the mean Nusselt Number at the left (a) up, and right (b) surface of the cylinder.
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
47
simulations are completely periodic (not presented here).
4.4. Variation of mean Nusselt number on cylinder surfaces with Reynolds numbers
Fig. 11. The variation of the mean local Nusselt number on the surface of the cylinder at z ¼ 3 (from average of 30 files).
vortex shedding, and the periods of the front and rear faces are half of the vortex shedding periods. The time history of the mean Nusselt numbers on cylinder surfaces obtained from 2-D
Fig. 10 compares the variation of mean Nusselt numbers on the cylinder walls for 2-D and 3-D simulations. On the front wall of the cylinder, the numerical predictions of 2-D and 3-D simulations are very close except at Re ¼ 220 where the flow conditions near the set of B-type instability. In this study, the mean Nusselt numbers obtained on the front face are slightly larger than that of Turki's (2003) result (Fig. 10(a)). However, 2-D simulations predict larger mean Nusselt numbers with a linear variation at the rear wall of the cylinder (Fig. 10(b)). At the rear wall of the cylinder, 3-D simulation yield a drop in the Nu values at Re ¼ 165 where the flow changes from 2-D to 3-D with A-type instability; on the other hand, it increases in the interval 165 Re ¼ 240. At the rear surface, the mean Nu values are larger than those of Turki's [12] at the 2-D flow region (Re < 165) and close to their prediction at 3-D flow region. At the upper surface, the mean Nusselt number slightly changes with Re number and 3-D simulation results are lower than those of the 2-D simulations and Turki's [12].
Fig. 12. The variation of the mean local Nusselt number on the cylinder surfaces along the spanwise direction at the midpoint of the surfaces (Re ¼ 185, from average of 30 files).
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N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
4.5. Variation of local Nusselt number
9.64 9.636
front surface
9.632 9.628 9.624 2.48 2.476 2.472 bottom surface
2.468 2.464 4.4 4 3.6 3.2 2.8 2.4 2.48
Nusselt Number
Nusselt Number
Nusselt Number
Nusselt Number
In Fig. 11, the variation of mean values of local Nusselt number on the cylinder walls are presented. The variation of mean values are computed by taking the average of 30 data files obtained at different times at the middle point of the cylinder z-axis (z ¼ 3). The largest Nusselt number values were obtained at the front wall. At the front stagnation point, these values are relatively small and take largest values at the end points of the front face. At the upper and bottom surfaces, the variation of mean Nusselt numbers are similar and takes the largest values at the edges which are close to rear surface, and the mean Nusselt number takes small values at the rear face. Fig. 11 also compares the local Nusselt number variations for Re ¼ 160, 185 and 250. For Re ¼ 160, variation local Nusselt number on the front surface is slightly underestimated than that provided by Sahu et al. [21] while the local Nusselt distributions on the other surfaces are observed to be identical. This deviation can be attributed to the choice of Prandtl numbers as Sahu's study [21] assumes Pr ¼ 1, this study uses Pr ¼ 0.7. For Re ¼ 250, the mean Nu number are larger on the front face and the effects of the Re number is much less on the other surfaces. In Figs. 12 and 13 illustrate the variations of the local Nusselt number on the cylinder walls in the streamwise direction for
Re ¼ 185 and 250 chosen representative of typical A and B type instability. The mean values are obtained at the mid-point of the cylinder at x-,y-axis. The mean local Nusselt number data presented are obtained by taking average of 30 data-files taken at different times on the walls of the cylinder. In Fig. 12, for Re ¼ 185, the variation of mean Nusselt number along z axis presents a deviation similar to sine curve. The maximum value of the Nusselt number is obtained at z ¼ 4.6 on the front, bottom and upper walls while it takes minimum value at z ¼ 2.65. However, the heat transfer rate becomes minimum at z ¼ 0.96 and maximum at z ¼ 2.4 on the rear face. There is 3.6D distance between the top points of the sinusoidal variation of the Nusselt number on front, upper and bottom faces while the same distances exist between the minimum points of Nusselt number variation on the rear surface. It is noticed that this distance is close to A-mode vortex formation wave length (z3D) along z-axis. The variation of the mean Nusselt number for Re ¼ 250, where B-mode exist in spanwise direction, is given at Fig. 13. Here, the variations of Nusselt numbers were irregular. The variation on upper and bottom surfaces are somewhat similar. The maximum heat transfer rate on the front surface take place at z ¼ 2.8 while it is at z ¼ 3.75 on the other faces. Deviation of the Nusselt number from its mean values in z-axis are very low on the front, upper and
rear surface u r
f
2.47 2.46
b
upper surface
2.45 2.44
0
2
4
6
z Fig. 13. The variation of the mean local Nusselt number on the cylinder surfaces along the spanwise direction at the midpoint of the surfaces (Re ¼ 250, from average of 30 files).
N. Mahir / International Journal of Thermal Sciences 119 (2017) 37e50
bottom surfaces; however, on the rear surface significant deviations occur. Fig. 14 presents instantaneous isotherms behind the cylinder corresponding to vortex structure given at Fig. 3(c) and instantaneous mean Nusselt number variation along z-axis. The isotherms appears dense at z ¼ 2.9 where maximum heat transfer rate take place while minimum heat transfer rates appears at z ¼ 1.2 and 4.8 with the coarse isotherm curves near the cylinder. For Re ¼ 250, instantaneous isotherms corresponds to Fig. 4(c) and variation of
Fig. 14. Close view of the isotherms at the rear face of the cylinder at the same instance with vortex structure shown in Fig. 3c (a) and corresponding local Nusselt number variation (b) (Re ¼ 185).
49
instantaneous local Nusselt number along z-axis is given in Fig. 15. At z ¼ 0.8, 2.4 and 3.8, the isotherms at the vicinity of the cylinder are more condensed, and the peaks at the mean Nusselt numbers distribution appears at these locations. The variation of the mean Nusselt number for Re ¼ 250, where B-mode exist in spanwise direction, is given at Fig. 13. Here, the variations of Nusselt numbers were irregular. 5. Conclusions In this study, three-dimensional unsteady flow and heat transfer from an isothermal square cylinder in crossflow of air is numerically investigated. The governing continuity, momentum and energy equations are solved numerically via implicit fractional step method where a third-order upwind scheme in first order derivatives is adapted. The heat transfer characteristics from a square cylinder in cross flow of air and the effects of 3D flow was assessed for Re ¼ 185 and 250. The study yield the following conclusion: In this study also A-mode and B-mode type flow regimes were identified by instantaneous vorticity lines at the wake of the cylinder and time history of the mean Nusselt numbers on the cylinder surfaces obtained. The isotherms and vorticity curves presents similarity in x-y plane while their view along z-axis differ. Time history of Nusselt number also differs on the cylinder faces with time delays between their maximum values on the cylinder faces. The time delay is more pronounced with 0.13Te2.1T at Re ¼ 185 where A-type flow mode exist in the wake. The variation of the mean Nusselt number along z-axis also differs depending on the flow regimes in spanwise direction. At Re ¼ 185 (A-mode flow), the distribution is approximately sinusoidal while it is less regular at Re ¼ 250 (B-mode). At both flow regimes, the amplitudes of the oscillations of Nusselt number are very small on front, up and down faces while they have relatively large amplitudes on the rear face. The heat transfer rates are larger where the temperature isosurfaces are more condensed which appear at the back face of the cylinder. At Re ¼ 250 (B-mode), the difference between these larger heat transfer locations vary as 1.4D-1.7D. Acknowledgment The author gratefully acknowledges the financial support by the Research Fund of Eskis¸ehir Osmangazi University, Project #201415039. References
Fig. 15. Close view of the isotherms at the rear surface of the cylinder at the same instance with vortex structure shown in Fig. 4c (a), and corresponding local Nusselt number variation (b) (Re ¼ 250).
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