ON THE ONSET OF DEAN VORTICES AND UNSTEADY SOLUTIONS THROUGH A CURVED RECTANGULAR DUCT OF LARGE ASPECT RATIO

Year 2015, Volume: 7 Issue: 2, 26 - 44, 01.06.2015

Poly Rani Shaha Sajal Kanti Rudro Nayan Kumar Poddar Rabindra Nath Mondal

https://doi.org/10.24107/ijeas.251251

Abstract

The present study covers a comprehensive numerical study on fully developed two-dimensional flow of viscous incompressible fluid through a curved rectangular duct of aspect ratio 4. Numerical calculations are carried out by using a spectral method, and covering a wide range of the Dean number,100 DnCase I: as formation of Dean vortices on unsteady solutions as the unsteady flow is steady-state, periodic, multi-periodic or chaotic, if the Dn is increased. Time evolution calculations show that the steady-state flow turns into chaotic flow through various flow instabilities, if Dn is increased no matter what the curvature is. It is found that the unsteady flow is a steady-state solution for small Dn`s, and it oscillates periodically or non-periodically (chaotic) between two-, four-, six-, eight- ten- and twelve-vortex solutions, if Dn is increased. In this study, we obtained multi-vortex solutions due to strong centrifugal force. It is found that the chaotic solution is weak for small Dn’s but strong as Dn becomes large. It is also found that the axial flow shifted to the outer wall of the duct as Dn increases

Keywords

Curved duct, secondary flow, unsteady solutions, Dean number, aspect ratio

References

• 1] Dean, W. R., Note on the motion of fluid in a curved pipe, Philos. Mag., 4, 208–223, 1927.
• [2] Berger, S.A., Talbot, L. and Yao, L. S., Flow in Curved Pipes, Annual. Rev. Fluid. Mech., 35, pp. 461-512, 1983.
• [3] Nandakumar, K. and Masliyah, J. H., Swirling Flow and Heat Transfer in Coiled and Twisted Pipes, Adv. Transport Process., 4, pp. 49-112, 1986.
• [4] Ito, H, Flow in Curved Pipes, JSME Int. J., 30, pp. 543-5, 1987.
• [5] Akiyama, M., Kikuchi, K., Nakayama, J., Nishiwaki, I. and Cheng, K. C., Hydraulically unstable flow, interactions of the boundary type secondary flows and their steplike development of transition, Trans. JSME, Ser. B47, 1705, 1981.
• [6] Ligrani, P. M. and Niver, R. D., Flow visualization of Dean vortices in a curved channel with 40 to 1 aspect ratio, Phys. Fluids, 31, 3605, 1988.
• [7] Yamamoto, K., Wu, X., Nozaki, K. and Hayamizu, Y., Visualization of Taylor–Dean flow in a curved duct of square cross-section, Fluid Dyn. Res., 38(1), 1-18, 2006.
• [8] Bara, B. M., Experimental investigation of developing and fully developed flow in a curved duct of square cross section, Ph.D. dissertation, University of Alberta, 1991.
• [9] Bara, B. M., Nandakumar, K. and Masliyah, J. H., An experimental and numerical study of the Dean problem: flow development towards two dimensional multiple solutions, J. Fluid Mech. 244, 339, 1992.
• [10] Baylis, J. A., Experiments on laminar flow in curved channels of square section, Journal of Fluid Mechanics, 48(3), 417-422, 1971.
• [11] Humphrey, J. A. C., Taylor, A. M .K., Whitelaw, Laminar flow in a square duct of strong curvature, Journal of Fluid Mechanics, 83, 509-527, 1977.
• [12] Cheng, K. C., Nakayama, J. and Akiyama, M., Effect of Finite and Infinite Aspect Ratios on Flow Patterns in Curved Rectangular Channels, in: Flow Visualization International Symposium, Tokyo, 181, 1977.
• [13] Ghia, K. N. and Sokhey, J. S., Laminar incompressible viscous flow in curved ducts of rectangular cross-section, Trans, ASME I: Journal of Fluids Engineering, 99, 640-648, 1977.
• [14] Sugiyama, T. Hayashi, T. and Yamazaki, K., Flow characteristics in the curved rectangular channels, Bulletin of JSME, 26(216), 532-552, 1983.
• [15] Chandratilleke, T. T. and Nursubyakto, Numerical prediction of secondary flow and convective heat transfer in externally heated curved rectangular ducts, Int. J. Thermal Sciences, 42, Issue 2, 187- 198, 2003.
• [16] Yanase, S. Mondal, R. N. and Kaga, Y., Numerical Study of Non-isothermal Flow with Convective Heat Transfer in a Curved Rectangular Duct, Int. J. Thermal Sciences, 44, 1047-1060, 2005.
• [17] Norouzi, M., Kayhani, M. H., Shu, C. and Nobari, M. R. H., Flow of second-order fluid in a curved duct with square cross-section, Journal of Non-Newtonian Fluid Mechanics, 165, 323–339, 2010.
• [18] Chandratilleke, T. T., Nadim, N. and Narayanaswamy, R., Vortex structure-based analysis of laminar flow behaviour and thermal characteristics in curved ducts, Int. J. Thermal Sciences, 59, 75- 86, 2012.
• [19] Norouzi, M. and Biglari, N., An analytical solution for Dean flow in curved ducts with rectangular cross section, Physics of Fluids, 25, 053602, 1-15, 2013.
• [20] Yanase, S. and Nishiyama, K., On the bifurcation of laminar flows through a curved rectangular tube, J. Phys. Soc. Japan, 57(11), 3790-3795, 1988.
• [21] Yanase, S., Kaga, Y. and Daikai, R., Laminar flow through a curved rectangular duct over a wide range of the aspect ratio, Fluid Dynamics Research, 31, 151-183, 2002.
• [22] Wang, L. and Liu, F., Forced convection in tightly coiled ducts: Bifurcation in a high Dean number region, International Journal of Non-Linear Mechanics, 42, 1018 – 1034, 2007.
• [23] Yanase, S., Mondal, R. N., Kaga, Y. and Yamamto, K.,Transition from steady to Chaotic States of Isothermal and Non-isothermal Flows through a curved Rectangular Duct, Journal of the Physics Society of Japan, 74(1), 345-358, 2005.
• [24] Mondal, R. N., Kaga, Y., Hyakutake, T. and Yanase, S., Effects of curvature and convective heat transfer in curved square duct flows, Trans. ASME, Journal of Fluids engineering, 128 (9), 1013—1023, 2006.
• [25] Mondal, R. N., Kaga, Y., Hyakutake, T. and Yanase, S., Bifurcation diagram for twodimensional steady flow and unsteady solutions in a curved square duct, Fluid Dynamics Research, 39, 413-446, 2007.
• [26] Mondal, R. N., Islam, S., Uddin, K. and Hossain, M. A., Effects of aspect ratio on unsteady solutions through curved duct flow, Applied Mathematics and Mechanics, 34(9), 1107–1122, 2013.
• [27] Gottlieb, D. and Orazag, S. A., Numerical Analysis of Spectral Methods, Society for Industrial and Applied Mathematics, Philadelphia, USA, 1977.
• [28] Mondal, R. N., Isothermal and Non-isothermal flows through curved duct with square and rectangular cross-section, Ph.D. Thesis, Department of Mechanical Engineering, Okayama University, Japan, 2006.
• [29] Yanase, S., Watanabe, T and Hyakutake. T., Traveling-wave solutions of the flow in a curvedsquare duct, Physics of Fluids, 20, 124101, 1-8, 2008.