# Unsteady Heat and Fluid Flow through a Curved Channel with Rectangular Cross-section for Several Cases of Aspect Ratio

• Samir Chandra Ray
• Rabindra Nath Mondal
Keywords: Curved rectangular duct, secondary flow, unsteady solutions, Dean number, Taylor number, Grashof number, time evolution.

### Abstract

In this paper, a comprehensive numerical study is presented for the fully developed two-dimensional flow of viscous incompressible fluid through a curved rectangular duct with different aspect ratios 2 and 3 for a constant curvature 14Î´=0.1"> . Unsteady solutions are obtained by using a spectral method and covering a wide range of Dean number 14100â‰¤Dnâ‰¤1000">  and the Grashof number 141000â‰¤Grâ‰¤2000"> . The outer wall of the duct is heated while the inner wall is cooled. The main concern of this study is to find out the unsteady flow behavior i.e whether the unsteady flow is steady-state, periodic, multi-periodic or chaotic, if the Dean number or the Grashof number is changed. For the aspect ratio 2, it is found that the unsteady flow is a steady-state solution for 14Dn=100">  and 14Gr=100, 500, 1500, 2000">  but periodic at 14Dn=100">  and 14Gr=1000. "> If the Dean number is increased i.e. at 14=500">  , it is found the unsteady flow is periodic at 14Gr=1000, 1500 ">  but chaotic at 14Gr=100, 500, 2000.">  If the Dean number is increased further i.e. at 14 Dn=1000"> , the unsteady flow becomes chaotic for any value of Gr in the range. For the aspect ratio 3, however, it is found that the unsteady flow is a steady-state solution for 14 Dn=100">  at 14 Gr=100 "> and 14 Gr=2000 ">  but periodic at 14Dn=100">  and 14Gr=500,1000,1500"> . If the Dean number is increased i.e. at Dn = 500 and 1000, the unsteady flow becomes chaotic for any value of Gr in the range. Contours of secondary flow patterns and temperature profiles are also obtained, and it is found that the unsteady flow consists of a single-, two-, three- , four-, five-, six-, seven- and eight-vortex solutions. It is also found that the chaotic flow enhances heat transfer more significantly than the steady-state or periodic solutions as the Dean number are increased.

### References

[1] Dean, W. R., and J. M. Hurst. "Note on the motion of fluid in a curved pipe." Philos Mag, 4:208-23
[2] Berger, S. A., L. Talbot, and L. S. Yao. "Flow in curved pipes." Annual review of fluid mechanics 15.1 (1983): 461 – 512.
[3] Ito, Hidesato. "Flow in curved pipes." JSME international journal 30.262 (1987): 543 – 552.
[4] Nandakumar, K. "Swirling flow and heat transfer in coiled and twisted pipes." Advances in transport processes 4 (1986): 49 – 112.
[5] Yanase, Shinichiro, Yoshito Kaga, and Ryuji Daikai. "Laminar flows through a curved rectangular duct over a wide range of the aspect ratio." Fluid Dynamics Research 31.3 (2002): 151.
[6] Zhang, J.S., Zhang, B.Z. and Jü, J. (2001) Fluid Flow in a Rotating Curved Rectangular Duct. International Journal of Heat and Fluid Flow, 22, 583-592.
[7] Winters, Keith H. "A bifurcation study of laminar flow in a curved tube of rectangular cross-section." Journal of Fluid Mechanics 180 (1987): 343 – 369.
[8] Mondal R. N., Kaga Y., Hyakutake, T and Yanase, S. "Unsteady solutions and the bifurcation diagram for the flow through a curved square duct", Fluids Dynamics Research 39 (2007): 413 – 446.
[9] Wang, Liqiu, and Tianliang Yang. "Periodic oscillation in curved duct flows." Physica D: Nonlinear Phenomena 200.3-4 (2005): 296 – 302.
[10] Yanase, Shinichiro, and Koji Nishiyama. "On the bifurcation of laminar flows through a curved rectangular tube." Journal of the Physical Society of Japan 57.11 (1988): 3790 – 3795.
[11] Wang, Liqiu, and Fang Liu. "Forced convection in tightly coiled ducts: Bifurcation in a high Dean number region." International Journal of Non-Linear Mechanics 42.8 (2007): 1018 – 1034.
[12] Mondal, R. N., Islam, S., Uddin, K., & Hossain, A. "Effects of aspect ratio on unsteady solutions through curved duct flow." Applied Mathematics and Mechanics 34.9 (2013): 1107 – 1122.
[13] Chandratilleke, Tilak T. "Numerical prediction of secondary flow and convective heat transfer in externally heated curved rectangular ducts." International Journal of Thermal Sciences 42.2 (2003): 187 – 198.
[14] Norouzi M, Kayhani M. H, Shu C, Nobari M. R. H. "Flow of second-order fluid in a curved duct with square cross section", J Non-Newtonian Fluid Mech 165(2010): 323 – 339.
[15] Chandratilleke T. T, Nadim N., Narayanaswamy R. "Vortex structure-based analysis of laminar flow behavior and thermal characteristics in curved ducts", Int J Thermal Sci, 59(2012): 75 – 86.
[16] Zhang L. J, Zhang W. P, Ming P. J. "Numerical simulation of secondary flow in a curved square duct", Applied Mathematics and Materials, 681(2014): 33 – 40.
[17] Mondal, R. N., Islam, M. Z. and Pervin, R. "Combined effects of centrifugal and coriolis instability of the flow through a rotating curved duct of small curvature", Procedia Engineering, 90(2014): 261 – 267.
[18]Yamamoto, K., Wu, X., Nozaki, K. and Hayamizu, Y. (2006) Visualization of Taylor-Dean Flow in a Curved Duct of Square Cross-Section. Fluid Dynamics Research, 38, 1-18
[19] Nobari, M.R.H., Nousha, A. and Damangir, E. (2009) A Numerical Investigation of Flow and Heat Transfer in Ro-tating U-Shaped Square Ducts. International Journal of Thermal Sciences, 48, 590-601.
[20] Li Y, Wang X, Yuan S, Tan S. K. "Flow Development in Curved Rectangular Ducts with Continuously Varying Curvature", Experimental Thermal and Fluid Science,75(2016): 1 – 15.
[21] Gottlieb D and Orazag SA. "Numerical Analysis of Spectral Methods", Society of Industrial and Applied Mathematics, Philadelphia, USA (1977).
[22] Keller H. B. "Lectures on Numerical Methods in Bifurcation Problems", Springer, Berlin (1987).
Published
2018-08-23
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