Time-Dependent Flow with Convective Heat Transfer through a Curved Square Duct with Large Pressure Gradient
Affiliation(s)
1 Department of Mathematics, Jagannath University, Dhaka, Bangladesh. 2 Additional Director (Education), Training Directorate, BGB Head Quarter, Pilkhana, Dhaka, Bangladesh.
1 Department of Mathematics, Jagannath University, Dhaka, Bangladesh. 2 Additional Director (Education), Training Directorate, BGB Head Quarter, Pilkhana, Dhaka, Bangladesh.
ABSTRACT
A numerical study is presented for the fully developed two-dimensional laminar flow of viscous incompressible fluid through a curved square duct for the constant curvature δ = 0.1. In this paper, a spectral-based computational algorithm is employed as the principal tool for the simulations, while a Chebyshev polynomial and collocation method as secondary tools. Numerical calculations are carried out over a wide range of the pressure gradient parameter, the Dean number, 100 ≤ Dn ≤ 3000 for the Grashof number, Gr, ranging from 100 to 2000. The outer wall of the duct is treated heated while the inner wall cooled, the top and bottom walls being adiabatic. The main concern of the present study is to find out the unsteady flow behavior i.e. whether the unsteady flow is steady-state, periodic, multi-periodic or chaotic, if Dn or Gr is increased. It is found that the unsteady flow is periodic for Dn = 1000 at Gr = 100 and 500 and at Dn = 2000, Gr = 2000 but steady-state otherwise. It is also found that for large values of Dn, for example Dn = 3000, the unsteady flow undergoes in the scenario “periodic→chaotic→periodic”, if Gr is increased. Typical contours of secondary flow patterns and temperature profiles are also obtained, and it is found that the unsteady flow consists of single-, two-, three- and four-vortex solutions. The present study also shows that there is a strong interaction between the heating-induced buoyancy force and the centrifugal force in a curved square passage that stimulates fluid mixing and consequently enhance heat transfer in the fluid.
A numerical study is presented for the fully developed two-dimensional laminar flow of viscous incompressible fluid through a curved square duct for the constant curvature δ = 0.1. In this paper, a spectral-based computational algorithm is employed as the principal tool for the simulations, while a Chebyshev polynomial and collocation method as secondary tools. Numerical calculations are carried out over a wide range of the pressure gradient parameter, the Dean number, 100 ≤ Dn ≤ 3000 for the Grashof number, Gr, ranging from 100 to 2000. The outer wall of the duct is treated heated while the inner wall cooled, the top and bottom walls being adiabatic. The main concern of the present study is to find out the unsteady flow behavior i.e. whether the unsteady flow is steady-state, periodic, multi-periodic or chaotic, if Dn or Gr is increased. It is found that the unsteady flow is periodic for Dn = 1000 at Gr = 100 and 500 and at Dn = 2000, Gr = 2000 but steady-state otherwise. It is also found that for large values of Dn, for example Dn = 3000, the unsteady flow undergoes in the scenario “periodic→chaotic→periodic”, if Gr is increased. Typical contours of secondary flow patterns and temperature profiles are also obtained, and it is found that the unsteady flow consists of single-, two-, three- and four-vortex solutions. The present study also shows that there is a strong interaction between the heating-induced buoyancy force and the centrifugal force in a curved square passage that stimulates fluid mixing and consequently enhance heat transfer in the fluid.
Cite this paper
Mondal, R. , Helal, M. , Shaha, P. and Poddar, N. (2015) Time-Dependent Flow with Convective Heat Transfer through a Curved Square Duct with Large Pressure Gradient. Open Journal of Fluid Dynamics, 5, 238-255. doi: 10.4236/ojfd.2015.53026.
Mondal, R. , Helal, M. , Shaha, P. and Poddar, N. (2015) Time-Dependent Flow with Convective Heat Transfer through a Curved Square Duct with Large Pressure Gradient. Open Journal of Fluid Dynamics, 5, 238-255. doi: 10.4236/ojfd.2015.53026.
References
[1] Dean, W.R. (1927) Note on the Motion of Fluid in a Curved Pipe. Philosophical Magazine, 4, 208-223.
http://dx.doi.org/10.1080/14786440708564324 [2] Berger, S.A., Talbot, L. and Yao, L.S. (1983) Flow in Curved Pipes. Annual Review of Fluid Mechanics, 35, 461-512.
http://dx.doi.org/10.1146/annurev.fl.15.010183.002333 [3] Nandakumar, K. and Masliyah, J.H. (1986) Swirling Flow and Heat Transfer in Coiled and Twisted Pipes. Advances in Transport Process, 4, 49-112. [4] Ito, H. (1987) Flow in Curved Pipes. JSME International Journal, 30, 543-552. [5] Winters, K.H. (1987) A Bifurcation Study of Laminar Flow in a Curved Tube of Rectangular Cross-Section. Journal of Fluid Mechanics, 180, 343-369.
http://dx.doi.org/10.1017/S0022112087001848 [6] Daskopoulos, P. and Lenhoff, A.M. (1989) Flow in Curved Ducts: Bifurcation Structure for Stationary Ducts. Journal of Fluid Mechanics, 203, 125-148.
http://dx.doi.org/10.1017/S0022112089001400 [7] Mondal, R.N. (2006) Isothermal and Non-Isothermal Flows through Curved Duct with Square and Rectangular Cross-Section. Ph.D. Thesis, Department of Mechanical and Systems Engineering, Okayama University, Japan. [8] Yanase, S. and Nishiyama, K. (1988) On the Bifurcation of Laminar Flows through a Curved Rectangular Tube. Journal of the Physical Society of Japan, 57, 3790-3795.
http://dx.doi.org/10.1143/JPSJ.57.3790 [9] Yanase, S., Kaga, Y. and Daikai, R. (2002) Laminar Flow through a Curved Rectangular Duct over a Wide Range of the Aspect Ratio. Fluid Dynamics Research, 31, 151-183.
http://dx.doi.org/10.1016/S0169-5983(02)00103-X [10] Wang, L. and Yang, T. (2005) Periodic Oscillation in Curved Duct Flows. Physica D, 200, 296-302.
http://dx.doi.org/10.1016/j.physd.2004.11.003 [11] Wang, L.Q. and Yang, T.L. (2004) Multiplicity and Stability of Convection in Curved Ducts: Review and Progress, Advances in Heat Transfer, 38, 203-256.
http://dx.doi.org/10.1016/s0065-2717(04)38004-4 [12] Yanase, S., Mondal, R.N., Kaga, Y. and Yamamoto, K. (2005) Transition from Steady to Chaotic States of Isothermal and Non-Isothermal Flows through a Curved Rectangular Duct. Journal of the Physical Society of Japan, 74, 345-358.
http://dx.doi.org/10.1143/JPSJ.74.345 [13] Yanase, S., Mondal, R.N. and Kaga, Y. (2005) Numerical Study of Non-Isothermal Flow with Convective Heat Transfer in a Curved Rectangular Duct. International Journal of Thermal Sciences, 44, 1047-1060.
http://dx.doi.org/10.1016/j.ijthermalsci.2005.03.013 [14] Mondal, R.N., Kaga, Y., Hyakutake, T. and Yanase, S. (2007) Bifurcation Diagram for Two-Dimensional Steady Flow and Unsteady Solutions in a Curved Square Duct. Fluid Dynamics Research, 39, 413-446.
http://dx.doi.org/10.1016/j.fluiddyn.2006.10.001 [15] Mondal, R.N., Uddin M.S. and Yanase, S. (2010) Numerical Prediction of Non-Isothermal Flow through a Curved Square Duct. International Journal of Fluid Mechanics Research, 37, 85-99.
http://dx.doi.org/10.1615/InterJFluidMechRes.v37.i1.60 [16] Chandratilleke, T.T. and Nursubyakto, S. (2003) Numerical Prediction of Secondary Flow and Convective Heat Transfer in Externally Heated Curved Rectangular Ducts. International Journal of Thermal Sciences, 42, 187-198.
http://dx.doi.org/10.1016/S1290-0729(02)00018-2 [17] Mondal, R.N., Kaga, Y., Hyakutake, T. and Yanase, S. (2006) Effects of Curvature and Convective Heat Transfer in Curved Square Duct Flows. Journal of Fluids Engineering, 128, 1013-1023.
http://dx.doi.org/10.1115/1.2236131 [18] Norouzi, M., Kayhani, M.H., Nobari, M.R.H. and Karimi Demneh, M. (2009) Convective Heat Transfer of Viscoelastic Flow in a Curved Duct, World Academy of Science, Engineering and Technology, 32, 327-333. [19] Norouzi, M., Kayhani, M.H., Shu, C. and Nobari, M.R.H. (2010) Flow of Second-Order Fluid in a Curved Duct with Square Cross-Section. Journal of Non-Newtonian Fluid Mechanics, 165, 323-339.
http://dx.doi.org/10.1016/j.jnnfm.2010.01.007 [20] Chandratilleke, T.T., Nadim, N. and Narayanaswamy, R. (2012) Vortex Structure-Based Analysis of Laminar Flow Behaviour and Thermal Characteristics in Curved Ducts. International Journal of Thermal Sciences, 59, 75-86.
http://dx.doi.org/10.1016/j.ijthermalsci.2012.04.014 [21] Gottlieb, D. and Orazag, S.A. (1977) Numerical Analysis of Spectral Methods. Society for Industrial and Applied Mathematics, Philadelphia. [22] Ligrani, P.M. and Niver, R.D. (1988) Flow Visualization of Dean Vortices in a Curved Channel with 40 to 1 Aspect Ratio. Physics of Fluids, 31, 3605.
http://dx.doi.org/10.1063/1.866877
[1] Dean, W.R. (1927) Note on the Motion of Fluid in a Curved Pipe. Philosophical Magazine, 4, 208-223.
http://dx.doi.org/10.1080/14786440708564324 [2] Berger, S.A., Talbot, L. and Yao, L.S. (1983) Flow in Curved Pipes. Annual Review of Fluid Mechanics, 35, 461-512.
http://dx.doi.org/10.1146/annurev.fl.15.010183.002333 [3] Nandakumar, K. and Masliyah, J.H. (1986) Swirling Flow and Heat Transfer in Coiled and Twisted Pipes. Advances in Transport Process, 4, 49-112. [4] Ito, H. (1987) Flow in Curved Pipes. JSME International Journal, 30, 543-552. [5] Winters, K.H. (1987) A Bifurcation Study of Laminar Flow in a Curved Tube of Rectangular Cross-Section. Journal of Fluid Mechanics, 180, 343-369.
http://dx.doi.org/10.1017/S0022112087001848 [6] Daskopoulos, P. and Lenhoff, A.M. (1989) Flow in Curved Ducts: Bifurcation Structure for Stationary Ducts. Journal of Fluid Mechanics, 203, 125-148.
http://dx.doi.org/10.1017/S0022112089001400 [7] Mondal, R.N. (2006) Isothermal and Non-Isothermal Flows through Curved Duct with Square and Rectangular Cross-Section. Ph.D. Thesis, Department of Mechanical and Systems Engineering, Okayama University, Japan. [8] Yanase, S. and Nishiyama, K. (1988) On the Bifurcation of Laminar Flows through a Curved Rectangular Tube. Journal of the Physical Society of Japan, 57, 3790-3795.
http://dx.doi.org/10.1143/JPSJ.57.3790 [9] Yanase, S., Kaga, Y. and Daikai, R. (2002) Laminar Flow through a Curved Rectangular Duct over a Wide Range of the Aspect Ratio. Fluid Dynamics Research, 31, 151-183.
http://dx.doi.org/10.1016/S0169-5983(02)00103-X [10] Wang, L. and Yang, T. (2005) Periodic Oscillation in Curved Duct Flows. Physica D, 200, 296-302.
http://dx.doi.org/10.1016/j.physd.2004.11.003 [11] Wang, L.Q. and Yang, T.L. (2004) Multiplicity and Stability of Convection in Curved Ducts: Review and Progress, Advances in Heat Transfer, 38, 203-256.
http://dx.doi.org/10.1016/s0065-2717(04)38004-4 [12] Yanase, S., Mondal, R.N., Kaga, Y. and Yamamoto, K. (2005) Transition from Steady to Chaotic States of Isothermal and Non-Isothermal Flows through a Curved Rectangular Duct. Journal of the Physical Society of Japan, 74, 345-358.
http://dx.doi.org/10.1143/JPSJ.74.345 [13] Yanase, S., Mondal, R.N. and Kaga, Y. (2005) Numerical Study of Non-Isothermal Flow with Convective Heat Transfer in a Curved Rectangular Duct. International Journal of Thermal Sciences, 44, 1047-1060.
http://dx.doi.org/10.1016/j.ijthermalsci.2005.03.013 [14] Mondal, R.N., Kaga, Y., Hyakutake, T. and Yanase, S. (2007) Bifurcation Diagram for Two-Dimensional Steady Flow and Unsteady Solutions in a Curved Square Duct. Fluid Dynamics Research, 39, 413-446.
http://dx.doi.org/10.1016/j.fluiddyn.2006.10.001 [15] Mondal, R.N., Uddin M.S. and Yanase, S. (2010) Numerical Prediction of Non-Isothermal Flow through a Curved Square Duct. International Journal of Fluid Mechanics Research, 37, 85-99.
http://dx.doi.org/10.1615/InterJFluidMechRes.v37.i1.60 [16] Chandratilleke, T.T. and Nursubyakto, S. (2003) Numerical Prediction of Secondary Flow and Convective Heat Transfer in Externally Heated Curved Rectangular Ducts. International Journal of Thermal Sciences, 42, 187-198.
http://dx.doi.org/10.1016/S1290-0729(02)00018-2 [17] Mondal, R.N., Kaga, Y., Hyakutake, T. and Yanase, S. (2006) Effects of Curvature and Convective Heat Transfer in Curved Square Duct Flows. Journal of Fluids Engineering, 128, 1013-1023.
http://dx.doi.org/10.1115/1.2236131 [18] Norouzi, M., Kayhani, M.H., Nobari, M.R.H. and Karimi Demneh, M. (2009) Convective Heat Transfer of Viscoelastic Flow in a Curved Duct, World Academy of Science, Engineering and Technology, 32, 327-333. [19] Norouzi, M., Kayhani, M.H., Shu, C. and Nobari, M.R.H. (2010) Flow of Second-Order Fluid in a Curved Duct with Square Cross-Section. Journal of Non-Newtonian Fluid Mechanics, 165, 323-339.
http://dx.doi.org/10.1016/j.jnnfm.2010.01.007 [20] Chandratilleke, T.T., Nadim, N. and Narayanaswamy, R. (2012) Vortex Structure-Based Analysis of Laminar Flow Behaviour and Thermal Characteristics in Curved Ducts. International Journal of Thermal Sciences, 59, 75-86.
http://dx.doi.org/10.1016/j.ijthermalsci.2012.04.014 [21] Gottlieb, D. and Orazag, S.A. (1977) Numerical Analysis of Spectral Methods. Society for Industrial and Applied Mathematics, Philadelphia. [22] Ligrani, P.M. and Niver, R.D. (1988) Flow Visualization of Dean Vortices in a Curved Channel with 40 to 1 Aspect Ratio. Physics of Fluids, 31, 3605.
http://dx.doi.org/10.1063/1.866877