Influence of rounding corners on unsteady flow and heat transfer around a square cylinder

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1 Influence of rounding corners on unsteady flow and heat transfer around a square cylinder S. K. Singh Deptt. of Mech. Engg., M. B. M. Engg. College / J. N. V. University, Jodhpur, Rajasthan, India Abstract A numerical study is performed to study the effect of rounding corners on flow and heat transfer characteristics around a square cylinder. Two square cylinders, one with sharp corners and other with rounded corners are considered. The rounded corners have a corners radius of d/4. Where, d is projected width of the square cylinder. The heated square cylinder is assumed to be horizontally placed in unconfined boundaries and flow of air (Pr = 0.7) is in vertical upward direction. The Reynolds number (Re) of the flow is 100. Numerical simulation results are presented in form of streamlines, vorticity contours, isotherm patterns, time history of lift and drag s, power spectra of lift s, Strouhal number, recirculation length, averaged drag, local Nusselt number and averaged Nusselt number. A comparison of the results for sharp corners and rounded corners square cylinders are presented. Results show that due to rounding corners of the square cylinder, flow separation becomes smooth and instability in the flow field is delayed. The size of the recirculation region as well as transverse extent of the wake decreases. Therefore, drag decreases due to reduced pressure drag. An enhancement in heat transfer from the cylinder is observed due to effective utilization of whole front face including rounded corners. Key words Heat transfer, Rounded corners, Square cylinder, Vortex shedding. I. INTRODUCTION Cylinders of different cross section e.g. circular and square finds many applications in engineering such as heat exchanger tubes, high-rise buildings, electronics cooling etc. When fluid flows over these cylinders, separation occurs from both sides of the cylinder and instability is produced in the flow field that cause periodic shedding of vortices from the cylinder. For a sharp corners square cylinder, the separation points are fixed. Sharma and Eswaran [1] studied the flow structure and heat transfer characteristics for an isolated sharp corners square cylinder. It is imperative that if sharp corners of the square cylinder are changed with the rounded corners, a drastic change in the flow and heat transfer characteristics of the cylinder can occur. Tamura et al.[2] studied the effect of corner shape on aerodynamic characteristics of the square cylinder. Dalton and Zheng [3] studied uniform flow past square and diamond cylinders with and without corner modifications. Hu et al. [4] experimentally studied the near wake of square cylinder with different corners radii. Park et al. [5] performed numerical analysis for critical Reynolds number with corner radius variation. Kumar et al. [6] studied near wake characteristics of transversely oscillating cylinders with different corner radii. The above mentioned studies involving corner modifications mainly deals with the effect of rounding corners on the aerodynamic behaviour. No systematic study is available in the literature that deals with the effect of rounding corners on flow structures and heat transfer from a square cylinder. Therefore, a numerical study is done to study the influence of rounding corners on flow and heat transfer characteristics of a square cylinder. One sharp corners and other rounded corners cylinder is modelled. The rounded corners cylinder have a corners radius of d/4. Where, d is projected width of the square cylinder. The heated cylinders are placed horizontally and flow of air (Pr = 0.7) is in vertical upward direction. The Reynolds number (Re) of the flow is 100. Numerical simulation results are presented by streamlines, vorticity contours, isotherm patterns, time history of lift and drag s, power spectra of lift s, Strouhal number, recirculation length, averaged drag, local Nusselt number and averaged Nusselt number. A comparison of the results for sharp corners and rounded corners square cylinders are presented. II. PROBLEM DEFINITION Fig. 1 shows a schematic diagram of physical model of the problem, flow geometry and boundaries of two-dimensional computational domain for rounded corners square cylinder. The cylinder has a projected width 'd' and a corner radius 'r'. 6

10 d 40 d Free slip boundary Free slip boundary 30 d International Journal on Mechanical Engineering and Robotics (IJMER) 30 d Outlet boundary y At the outlet: = 0 At the vertical left and right

2 10 d 40 d Free slip boundary Free slip boundary 30 d International Journal on Mechanical Engineering and Robotics (IJMER) 30 d Outlet boundary y At the outlet: = 0 At the vertical left and right boundaries, U = 0, V = 1 and θ = 0 i.e. free slip boundary conditions. At the cylinder surface, U = V = 0 and θ = 1 i.e. no slip condition. The flow is assumed to start from the rest. IV. NUMERICAL METHODOLOGY r Fig.1: Schematic diagram of physical model and computational domain for rounded corners square cylinder. III. GOVERNING EQUATIONS The fluid flow is assumed to be two-dimensional unsteady laminar and incompressible. The governing equations are expressed in the following form: Continuity equation: (1) X- Momentum equation: d Inlet boundary Flow x (2) Numerical study is carried out by using control volume based CFD package FLUENT. The grid for the computational domain has been generated in GAMBIT. Fig. 2 shows the grid for rounded corners square cylinder. The grid independence study has been carried out to obtain results that are grid independent. The grids for sharp corners and rounded corners square cylinders were made having total number of 1,28,664 and 1,27,114 mixed cells, respectively. A dimensionless time step of size 0.01 is used in the computations. V. VALIDATION OF NUMERICAL RESULTS The results of present numerical computations have been validated by comparing with the results for sharp corners square cylinder available in the literature. Table I shows a comparison of mean drag, Strouhal number and averaged Nusselt number for the present work with the results of Sharma and Eswaran [1]. An excellent agreement is observed between the data of present numerical computations with that of Sharma and Eswaran [1]. This validates the grid used and the numerical computations. A maximum deviation of 0.3% in mean drag, 0.8% in averaged Nusselt number and 0.8% in Strouhal number is observed. Y- Momentum equation: (3) Energy equation: (4) where U = Free stream velocity, T = uniform temperature at inlet, T w = constant cylinder wall temperature. Fig.2: Grid for rounded corners square cylinder. Following boundary conditions are used: At the inlet: U = 0, V = 1 and θ = 0, 7

TABLE I: COMPARISON OF THE RESULTS FOR SHARP CORNERS SQUARE CYLINDER WITH LITERATURE. S. No Reference 1. Sharma and Eswaran [1] 2.

3: Instantaneous streamlines at two time instants in a vortex shedding cycle for sharp corners and rounded corners square cylinders. VI.

These results corresponds to the simulations for at least 10 vortex shedding cycles after the asymptotic vortex shedding frequency was attained. A. Instantaneous Streamlines Fig.

For both the cylinders, streamlines oscillate in the transverse direction at downstream locations. This shows presence of vortex shedding in the flow field.

Similarly for rounded corners cylinder, at t = 0, the streamlines touches left rounded corner of the front face and at t = T/2, the streamlines touches right rounded corner of the front face.

However, for rounded corners cylinder, separation occurs at the two rounded corners of the front face. B. Instantaneous Vorticity Contours Fig.

For both the cylinders, the vortices are formed and shed Fig.4: Instantaneous vorticity contours at two time instants in a vortex shedding cycle for sharp corners and rounded corners square cylinders.

The figure shows that due to rounding corners of the cylinder, the vortices come closer to the cylinder and interacts with opposite side vortices at a shorter distance from the cylinder.

3 TABLE I: COMPARISON OF THE RESULTS FOR SHARP CORNERS SQUARE CYLINDER WITH LITERATURE. S. No Reference 1. Sharma and Eswaran [1] 2. Present work Mean drag (C d ) Struohal number (St) Averaged Nusselt number (Nu avg ) Fig.3: Instantaneous streamlines at two time instants in a vortex shedding cycle for sharp corners and rounded corners square cylinders. VI. RESULTS AND DISCUSSION In this section, unsteady flow and heat transfer characteristics for sharp corners and rounded corners square cylinders are presented. These results corresponds to the simulations for at least 10 vortex shedding cycles after the asymptotic vortex shedding frequency was attained. A. Instantaneous Streamlines Fig. 3 shows instantaneous streamlines for sharp corners and rounded corners square cylinders at two time instants (t = 0 and T/2) in a vortex shedding cycle having a time period T. For both the cylinders, streamlines oscillate in the transverse direction at downstream locations. This shows presence of vortex shedding in the flow field. For sharp corners cylinder, at t = 0, the streamlines touches left corner of the front face and at t = T/2, the streamlines touches right corner of the front face. Similarly for rounded corners cylinder, at t = 0, the streamlines touches left rounded corner of the front face and at t = T/2, the streamlines touches right rounded corner of the front face. This indicates alternate behaviour of vortex shedding process for both the cylinders. For sharp corners cylinder, the flow is separated from two sharp corners of the front face. However, for rounded corners cylinder, separation occurs at the two rounded corners of the front face. B. Instantaneous Vorticity Contours Fig. 4 shows instantaneous vorticity contours for sharp corners and rounded corners square cylinders at two time instants (t = 0 and T/2) in a vortex shedding cycle (time period T). For both the cylinders, the vortices are formed and shed Fig.4: Instantaneous vorticity contours at two time instants in a vortex shedding cycle for sharp corners and rounded corners square cylinders. from left and right sides of the cylinder, alternatively. The figure shows that due to rounding corners of the cylinder, the vortices come closer to the cylinder and interacts with opposite side vortices at a shorter distance from the cylinder. Along the streamwise flow direction, the distance between shed vortices is also decreased due to rounding corners of the cylinder. C. Instantaneous Isotherm Patterns Fig. 5 show instantaneous isotherm patterns for sharp corners and rounded corners square cylinders for two time instants (t = 0 and T/2) in a vortex shedding cycle with time period T. For both the cylinders, instantaneous isotherms are also of oscillating nature. This shows unsteady periodic nature of the temperature field and existence of vortex shedding. The figure shows that due to rounding corners of the cylinder, isotherm patterns looks little compressed toward the cylinder. D. Time Averaged Isotherms Fig. 6 show time-averaged isotherms for sharp corners and rounded corners square cylinders. These are obtained from instantaneous isotherms averaged over 10 vortex shedding cycles. Time-averaged isotherms give an idea of heat transfer characteristics of the cylinder. There is 8

clustering of isotherms at the corners of the front face for sharp corners cylinder which signifies large heat transfer at the front corners.

181 and 0.178 for the sharp corners and rounded corners square cylinders, respectively.

This indicates that rounding corners of the cylinder stabilizes the flow and instability produced in the flow field is delayed. Fig.

6: Time averaged isotherms for sharp corners and rounded corners square cylinders. isotherms is seen at the whole front face including the rounded corners of the cylinder.

4 clustering of isotherms at the corners of the front face for sharp corners cylinder which signifies large heat transfer at the front corners. For rounded corners cylinder, the clustering of marginal decrease in the amplitude of oscillations due to rounding corners of the square cylinder. The r.m.s. value of oscillating lift (C lrms ) is and for the sharp corners and rounded corners square cylinders, respectively. Comparison of figures 7 (a) and (b) show that oscillations in the lift start early for the sharp corners cylinder as compared to the rounded corners cylinder. This indicates that rounding corners of the cylinder stabilizes the flow and instability produced in the flow field is delayed. Fig.5: Instantaneous isotherms at two time instants in a vortex shedding cycle for sharp corners and rounded corners square cylinders. (a) Fig.6: Time averaged isotherms for sharp corners and rounded corners square cylinders. isotherms is seen at the whole front face including the rounded corners of the cylinder. This signifies large heat transfer from the whole front face. (b) E. Time History of Lift Coefficient Lift is defined as ratio of the lift force produced around the cylinder to the dynamic force on projected area of the cylinder. Fig. 7 (a) and (b) show the time history of lift for sharp corners and rounded corners square cylinders, respectively for complete duration of the simulation. For both cylinders, the variation of lift is a horizontal straight line initially. After some time, lift starts oscillating due to instability produced in the flow field. The amplitude of oscillating lift increases with time and finally reaches to a maximum value. The nature of lift is unsteady periodic due to presence of vortex shedding. Fig. 7 (c) shows a comparison of lift for sharp corners and rounded corners square cylinders on magnified view. The figure shows that variation of lift for both the cylinders is a pure sine wave oscillating about mean zero value. However, there is a (c) Fig.7: Time history of lift for (a) sharp corners (complete), (b) rounded corners (complete), and (c) comparison on magnified view. F. Time History of Drag Coefficient Drag is defined as ratio of the drag force acting on the cylinder to the dynamic force on projected area of the cylinder. Fig. 8 (a) and (b) show the time history of drag for sharp corners and rounded corners 9

5 square cylinders, respectively. For both the cylinders, initially, the drag sharply decreases from a high value to a minimum value. It then sharply increases and starts oscillating which finally reaches to a constant amplitude of oscillation about a mean value. This shows that the drag force is also of unsteady periodic nature. Comparison of figures 8 (a) and (b) show that mean value of oscillating drag becomes lower due to rounding corners of the cylinder and shape of the curve at bottom becomes little wider. G. Power Spectra of Lift Coefficient Fig. 9 shows a comparison of Power spectra of lift for sharp corners and rounded corners square cylinders. Power spectra has been calculated from Fast Fig. 9. Comparison of Power spectra of lift for sharp corners and rounded corners square cylinders. H. Strouhal Number Strouhal number is non-dimensional vortex shedding frequency (f) given as, St = fd/u. Table II shows a comparison of Strouhal number for sharp corners and rounded corners square cylinders. Due to increase in vortex shedding frequency, the value of strouhal number is higher for rounded corners cylinder as compared to the sharp corners TABLE III: COMPARISON OF STROUHAL NUMBER (a) Geometry Struohal number (St) Sharp corners Rounded corners cylinder. This indicates that vortex structures in the near wake forms and sheds at a faster rate for rounded corners cylinder. I. Streamwise Mean Velocity and Recirculation Length (b) Fig.8: Time history of drag for (a) sharp corners, and (b) rounded corners square cylinders. Fourier Transform (FFT) of the time history of lift. A sharp peak is seen in the spectra for both the cylinders at the frequency to the vortex shedding. The spectral power of the rounded corners cylinder is slightly less as compared to that of the sharp corners cylinder due to marginal decrease in the oscillating amplitude of lift. However, peak in the spectra shifts to a higher value of frequency. This shows a increase in vortex shedding frequency due to rounding corners of the cylinder. Fig. 10 shows the distribution of non-dimensionalized streamwise mean velocity along the wake centerline for sharp corners and rounded corners square cylinders. A region of reversed flow with negative mean velocity is seen near base of both the cylinders. The mean velocity first decreases to a maximum negative value and then increases to positive values and finally reaches to asymptotic constant value at around y/d = 10 for both the cylinders. However, the location of maximum negative mean velocity shifts towards the base of the cylinder due to rounding corners of the cylinder. Mean recirculation length (L r ) is defined as a non-dimensional distance in streamwise direction from surface of the cylinder to the point where mean velocity becomes zero along the wake centerline. Mean recirculation length has been calculated as 1.98 and 1.75 for the sharp corners and rounded corners square cylinders, respectively. This shows that size of the recirculation region decreases due to rounding corners of the cylinder. 10

6 J. Averaged Drag Coefficient Fig. 8 shows that drag oscillates about a mean value due to alternate periodic vortex shedding in the flow field. Averaged drag (C d ) is calculated by averaging the oscillating drag for at least 10 cycles of vortex shedding after peak amplitude of oscillations have been reached. Averaged value of total drag is the sum of the pressure drag and viscous drag. Table III shows a comparison of pressure drag, viscous drag and total drag s for sharp corners and rounded corners square cylinders. The table shows that due to rounding corners of the cylinder, viscous drag increases and pressure drag decreases. For sharp corners cylinder, the separating shear layers are suddenly displaced away from the cylinder after separation from the sharp corners. This creates a wider wake, which results in a higher pressure drag. However, for rounded corners cylinder, the flow separates smoothly from the rounded corners and transverse extent of the wake becomes smaller resulting in a lower pressure drag. But, viscous drag becomes higher due to flow gliding along the rounded corners. The overall effect due to rounding corners of the cylinder is a decrease of about 8% in the total drag. K. Local Nusselt Number Sharp corners Rounded corners In the present study, local Nusselt number on the cylinder surface is obtained by averaging the oscillating Nusselt number for at least 10 cycles of vortex shedding after peak amplitude of oscillations have been reached. Fig.11 compares the variation of local Nusselt number on the cylinder surface for sharp corners and rounded corners square cylinders. For both the cylinders, local Nusselt number is highest on the front surface. This show that highest heat transfer takes place from the front surface as compared to the other surfaces. For sharp corners cylinder, local Nusselt number shows highest sharp peaks at the front corners points, B and C. However, for rounded corners cylinder, highest rounded peaks are observed at the front separation points, d and g. The figure shows that due to rounding corners of the cylinder, the local Nusselt number is smoothly distributed over the front rounded corners instead of sharp peaks. At the front face, minimum value of Nusselt number is observed at the mid points for both the cylinders. However, minimum value of Nu at the front face is higher for the rounded corners cylinder. The lowest value of Nusselt number is observed at the rear face of both the cylinders. Local Nusselt number (Nu) is used to express heat transfer from the cylinder surface to the fluid. It is given by Nu = hd/k. Where h is the local heat transfer on the cylinder surface and k is thermal conductivity of the fluid. Fig. 10. Distribution of streamwise mean velocity along the wake center-line for sharp corners and rounded corners square cylinders. TABLE III: COMPARISON OF PRESSURE DRAG, VISCOUS DRAG AND TOTAL DRAG COEFFICIENTS Geometry Pressure drag Viscous drag Total drag (C d ) Fig. 11. Variation of local Nusselt number on the surfaces of sharp corners and rounded corners square cylinders. L. Averaged Nusselt Number To calculate the change in heat transfer due to rounding corners of the cylinder, a surface averaged Nusselt number (Nu avg ) is calculated by averaging the local Nusselt number (Nu) over the entire cylinder surface. Table IV shows a comparison of the surface averaged Nusselt number for sharp corners and rounded corners square cylinders. Table shows that surface averaged Nusselt number is higher for rounded corners cylinder as compared to the sharp corners cylinder. An enhancement in the average value of Nusselt number is about 17% due to rounding corners of the cylinder. This enhancement can be attributed to effective utilization of whole front face with rounded corners of the square cylinder for heat transfer. (C dp ) (C dv ) 11

7 TABLE IV: COMPARISON OF AVERAGED NUSSELT NUMBER Geometry Averaged Nusselt number (Nu avg ) Sharp corners Rounded corners VII. CONCLUSION A numerical study on the effect of rounding corners on unsteady flow and heat transfer around a square cylinder is presented in this paper. Two-dimensional numerical simulations are carried out at Re = 100 for sharp corners and rounded corners square cylinders. Comparison of the results show that due to rounding corners of the square cylinder, a marginal decrease in lift is observed. However, 8% reduction is drag is observed. Length of recirculation region decreases and Strouhal number increases. About 17% enhancement in heat transfer is observed due to rounding corners of the square cylinder. ACKNOWLEDGMENT The author is thankful to Dr. Rohit Misra, G.E.C., Ajmer for providing necessary support to perform this work. REFERENCES [1] A. Sharma, and V. Eswaran, "Heat and fluid flow across a square cylinder in the two-dimensional laminar flow regime," Numerical Heat Transfer, Part A, 45, pp , [2] T. Tamura, T. Miyagi, and T. Kitagishi, "Numerical prediction of unsteady pressures on a square cylinder with various corner shapes," J. Wind Engg. and Ind. Aerodyn., vol , pp , [3] C. Dalton, and W. Zheng, "Numerical solutions of a viscous uniform approach flow past square and diamond cylinders," J. Fluids and Struct., vol. 18, pp , [4] J. C. Hu, Y. Zhou, and C. Dalton, Effects of the corner radius on the near wake of a square prism, Experiments in Fluids, vol.40, no.1, pp , [5] D. Park, K. Yang, K. Lee, and C. Kang, "Effects of rounding corners on the flow past a square cylinder," J. comput. Fluids Eng., vol. 19, no. 1, pp , 2014 (in Korean). [6] R. A. Kumar, C. H. Sohn, and B. H. L. Gowda, "A PIV study of the near wake flow features of a square cylinder: influence of corner radius," Journal of Mechanical Science and Technology, vol. 29, no.2, ,

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