Theoretical and Computational Fluid Dynamics

, Volume 31, Issue 2, pp 159–187 | Cite as

Numerical investigation of mixed convection heat transfer from two isothermal circular cylinders in tandem arrangement: buoyancy, spacing ratio, and confinement effects

  • Erick Salcedo
  • Juan C. Cajas
  • César Treviño
  • Lorenzo Martínez-Suástegui
Original Article

Abstract

This paper presents a two-dimensional numerical study for mixed convection in a laminar cross-flow with a pair of stationary equal-sized isothermal cylinders in tandem arrangement confined in a channel. The governing equations are solved using the control volume method on a nonuniform orthogonal Cartesian grid, and the immersed boundary method is employed to identify the cylinders placed in the flow field. The numerical scheme is first validated against standard cases of symmetrically confined isothermal circular cylinders in plane channels, and grid convergence tests were also examined. The objective of the present study was to investigate the influence of buoyancy and the blockage ratio constraint on the flow and heat transfer characteristics of the immersed cylinder array. Using a fixed Reynolds number based on cylinder diameter of \(Re_{D} = 200\), a fixed value of the Prandtl number of \(Pr = 7\), and a blockage ratio of \(D/H = 0.2\), all possible flow regimes are considered by setting the longitudinal spacing ratio (\(\sigma = L/D\)) between the cylinder axes to 2, 3, and 5 for values of the buoyancy parameter (Richardson number) in the range \(-1\le Ri\le 4\). The interference effects and complex flow features are presented in the form of mean and instantaneous velocity, vorticity, and temperature distributions. The results demonstrate how the buoyancy, spacing ratio, and wall confinement affect the wake structure and vortex dynamics. In addition, local and average heat transfer characteristics of both cylinders are comprehensively presented for a wide range in the parametric space.

Keywords

Mixed convection Tandem cylinders Blockage ratio Interference effects Bimodal vortex shedding 

Abbreviations

BR

Blockage ratio, D/H

D

Cylinder diameter (characteristic length)

f

Vortex shedding frequency (Hz)

g

Gravity acceleration

Gr

Grashof number based on cylinder diameter, \(Gr = \hbox {g}\beta (T_{w}-T_{0})D^{3}/\nu ^{2}\)

h

Local heat transfer coefficient

H

Width of computational domain

k

Thermal conductivity of fluid

L

Pitch (center-to-center distance between two cylinders)

\(L_{v1}/D\)

Wake closure length of the upstream cylinder

\(L_{v2}/D\)

Wake closure length of the downstream cylinder

\(L_\mathrm{tot}\)

Length of computational domain

n

Normal direction

Nu

Local Nusselt number (see Eq. 8)

\(\overline{Nu}\)

Average Nusselt number (see Eq. 9)

Pe

Peclet number, \(Pe = u_{0}D/\alpha \)

Pr

Prandtl number, \(Pr = \nu /\alpha \)

\(Re_{D}\)

Reynolds number based on \(u_D\), \(Re_{D} = u_DD/\nu \)

Ri

Richardson number based on cylinder diameter, \(Ri = Gr/Re^{2}\)

S

Length from the origin to the channel outlet

SD

Standard deviation

St

Strouhal number based on cylinder diameter, \(St = fD/u_{0}\)

t

Time

T

Temperature

\(T_{0}\)

Fluid temperature at the channel inlet

\(T_{w}\)

Temperature at the cylinders’ surface

\(u_{0}\)

Fluid velocity at the channel inlet

uv

Longitudinal and transverse velocity components, respectively

\(u_{D}\)

Average longitudinal velocity over the cylinders (see Eq. 10)

U

Nondimensional longitudinal velocity component, \(U = u/u_{0}\)

V

Nondimensional transverse velocity component, \(V = v/u_{0}\)

xy

Cartesian rectangular coordinates

X

Nondimensional longitudinal coordinate, \(X = x/D\)

Y

Nondimensional transverse coordinate, \(Y = y/D\)

Greek symbols

\(\alpha \)

Thermal diffusivity of fluid

\(\beta \)

Thermal volumetric expansion coefficient

\(\gamma _{s}\)

Separation angle

\(\mu \)

Dynamic viscosity

\(\nu \)

Kinematic viscosity

\(\psi \)

Nondimensional stream function

\(\Omega \)

Nondimensional vorticity

\(\sigma \)

Nondimensional pitch-to-diameter ratio, \(\sigma = L/D\)

\(\sigma _{v1}\)

Nondimensional wake closure length, \(\sigma _{v1} = L_{v1}/D\)

\(\sigma _{v2}\)

Nondimensional wake closure length, \(\sigma _{v2} = L_{v2}/D\)

\(\theta \)

Nondimensional temperature, \(\theta =(T-T_{0})/(T_{w}-T_{0})\)

\(\tau \)

Nondimensional time

Subscripts

0

Ambient or reference

1, 2

Refers to the upstream and downstream cylinder, respectively

w

At the surface of the cylinders

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alam, M., Meyer, J.: Two interacting cylinders in cross flow. Phys. Rev. E 84, 056304 (2011)CrossRefGoogle Scholar
  2. 2.
    Blevins, R.: Flow-Induced Vibration. Van Nostrand Reinhold, New York (1977)MATHGoogle Scholar
  3. 3.
    Buyruk, E.: Numerical study on heat transfer charactersitics on tandem cylinders, inline and staggered tube banks in cross- flow of air. Int. Commun. Heat Mass 29, 355–366 (2002)CrossRefGoogle Scholar
  4. 4.
    Chatterjee, D., Biswas, G., Amiroudine, S.: Mixed convection heat transfer from an in-line row of square cylinders in cross-flow at low Reynolds number. Numer. Heat Transf. A 61, 891–911 (2012)Google Scholar
  5. 5.
    Chatterjee, D., Amiroudine, S.: Two-dimensional mixed convection heat transfer from confined tandem square cylinders in cross-flow at low Reynolds numbers. Int. Commun. Heat Mass. 37, 7–16 (2010)CrossRefGoogle Scholar
  6. 6.
    Chatterjee, D., Mondal, B.: Mixed convection heat transfer from tandem square cylinders for various gap to size ratios. Numer. Heat Transf. A 63, 101–119 (2013)CrossRefGoogle Scholar
  7. 7.
    Chatterjee, D., Raja, M.: Mixed convection heat transfer past in-line square cylinders in a vertical duct. Therm. Sci. 17, 567–580 (2013)CrossRefGoogle Scholar
  8. 8.
    Chen, S.: Flow-Induced Vibration of Circular Cylindrical Structures. Hemisphere Publishing Coorporation, New York (1987)Google Scholar
  9. 9.
    Chen, J., Pritchard, W., Tavener, S.: Bifurcation of flow past a cylinder between parallel plates. J. Fluid Mech. 284, 23–41 (1995)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, C., Wang, T.S.: Finite analytic solution of convective heat transfer for tube arrays in cross flow: Part ii—heat transfer analysis. ASME J. Heat Transf. 111, 641–648 (1989)CrossRefGoogle Scholar
  11. 11.
    Dehkordi, B., Moghaddam, H., Jafari, H.: Numerical simulation of flow over two circular cylinders in tandem arrangement. J. Hydrodyn. 23, 114–126 (2011)CrossRefGoogle Scholar
  12. 12.
    Farramt, T., Tan, M., Price, W.: A cell boundary element method applied to laminar vortex-shedding from arrays of cylinders in various arrangements. J. Fluid Struct. 14, 375–402 (2000)CrossRefGoogle Scholar
  13. 13.
    Goharrizi, A., Sadeghi, R.: Thermophoretic deposition of aerosol particles in laminar mixed-convection flow in a channel with two heated built-in square cylinders. Adv. Powder Technol. 21, 320–325 (2010)CrossRefGoogle Scholar
  14. 14.
    Gori, F., Petracci, I., Tedesco, V.: Cooling of two smooth cylinders in row by a slot jet of air with low turbulence. Appl. Therm. Eng. 27, 2415–2425 (2007)CrossRefGoogle Scholar
  15. 15.
    Guillén, I., Treviño, C., Martínez-Suástegui, L.: Unsteady laminar mixed convection heat transfer from a horizontal isothermal cylinder in contra-flow: Buoyancy and wall proximity effects on teh flow response and wake structure. Exp. Therm. Fluid Sci. 52, 30–46 (2014)CrossRefGoogle Scholar
  16. 16.
    Harichandan, A., Roy, A.: Numerical investigation of flow past a single and tandem cylindrical bodies in the vicinity of a plane wall. J. Fluids Struct. 33, 19–43 (2012)CrossRefGoogle Scholar
  17. 17.
    Harimi, I., Saghafian, M.: Numerical simulation of fluid flow and forced convection heat transfer from tandem circular cylinders using overset grid method. J. Fluid Struct. 28, 309–327 (2012)CrossRefGoogle Scholar
  18. 18.
    Hetz, A.A., Dhaubhadel, M.N., Telionis, D.P.: Vortex shedding over five in-line cylinders cylinders. J. Fluid. Struct. 5, 243–257 (1991)CrossRefGoogle Scholar
  19. 19.
    Hiwada, M., Mabuchi, I., Yanagihara, H.: Fluid flow and heat transfer around two circular cylinders in cross flow. Bull. JSME 25, 1737–1745 (1982)CrossRefGoogle Scholar
  20. 20.
    Huang, Z., Xi, G., Zhang, W., Wen, S.: Mixed convection heat transfer from confined tandem square cylinders in a horizontal channel. Int. J. Heat Mass Transf. 66, 625–631 (2013)CrossRefGoogle Scholar
  21. 21.
    Igarashi, T.: Characteristics of the flow around two circular cylinders arranged in tandem, 1st report. Bull. JSME B 24, 323–331 (1981)CrossRefGoogle Scholar
  22. 22.
    Igarashi, T.: Characteristics of the flow around two circular cylinders arranged in tandem, 2nd report. Bull. JSME B 27, 2380–2387 (1984)CrossRefGoogle Scholar
  23. 23.
    Ishigai, S., Nishikawa, E., Nishimura, K., Cho, K.: Experimental study of structure of gas flow in tube banks with tube axes normal to flow (part 1, kármán vortex flow from two tubes at various spacings. Bull JSME 15, 949–956 (1972)CrossRefGoogle Scholar
  24. 24.
    Jiang, R.J., Lin, J.Z.: Wall effects on flows past two tandem cylinders of different diameters. J. Hydrodyn. 24, 1–10 (2012)CrossRefGoogle Scholar
  25. 25.
    Juncu, G.: A numerical study of momentum and forced convection heat transfer around two tandem circular cylinders at low Reynolds numbers. Momentum transfer. Int. J. Heat Mass Transf. 50, 3788–3798 (2007)CrossRefMATHGoogle Scholar
  26. 26.
    Kaptan, Y., Buyruk, E., Ecder, A.: Numerical investigation of fouling on cross-flow heat exchanger tubes with conjugated heat transfer approach. Int. Commun. Heat Mass 35, 1153–1158 (2008)CrossRefGoogle Scholar
  27. 27.
    Khan, W., Culham, J., Yovanovich, M.: Fluid flow and heat transfer from a cylinder between parallel planes. J. Thermophys. Heat Transf. 18, 395–403 (2004)CrossRefGoogle Scholar
  28. 28.
    Lee, T., Basu, S.: Nonintrusive measurements of the boundary layer developing on a single and two circular cylinders. Exp. Fluids 23, 187–192 (1997)CrossRefGoogle Scholar
  29. 29.
    Li, J., Chabbarel, A., Donneaud, M., Martin, R.: Numerical study of laminar flow past one and two cylinders. Comput. Fluids 19, 155–170 (1991)CrossRefMATHGoogle Scholar
  30. 30.
    Lin, J., Yang, Y., Rockwell, D.: Flow past two cylinders in tandem: instantaneous and averaged flow structure. J. Fluid. Struct. 16, 1050–1071 (2002)Google Scholar
  31. 31.
    Lin, J., Jiang, R., Chen, Z., Ku, X.: Poiseuille flow-induced vibrations of two cylinders in tandem. J. Fluid Struct. 40, 70–85 (2013)CrossRefGoogle Scholar
  32. 32.
    Liu, M.M., Lu, L., Teng, B., Zhao, M., Tang, G.Q.: Re-examination of laminar flow over twin circular cylinders in tandem arrangement. Fluid Dyn. Res. 46, 025501 (2014)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ljungkrona, L., Norberg, C., Sundén, B.: Free-stream turbulence and tube spacing effects on surface pressure fluctuations for two tubes in an in-line arrangement. J. Fluid. Struct. 5, 701–727 (1991)CrossRefGoogle Scholar
  34. 34.
    Ljungkrona, L., Sundén, B.: Flow visualization and surface pressure measurement on two tubes in an inline arrangement. Exp. Therm. Fluid Sci. 6, 15–27 (1993)CrossRefGoogle Scholar
  35. 35.
    Lu, J., Han, H., Shi, B.: A numerical study of fluid flow passes two heated/coled square cylinders in a tandem arrangement via lattice boltzmann method. Int. J. Heat Mass Transf. 55, 3909–3920 (2012)CrossRefGoogle Scholar
  36. 36.
    Mahir, N., Altaç, Z.: Numerical investigation of convective heat transfer in unsteady flow past two cylinders in tandem arrangements. Int. J. Heat Fluid Flow 29, 1309–1318 (2008)CrossRefGoogle Scholar
  37. 37.
    Mandhani, V., Chhaabra, R., Eswaran, V.: Forced convection heat transfer in tube banks in cross flow. Chem. Eng. Sci. 57, 379–391 (2002)CrossRefGoogle Scholar
  38. 38.
    Martínez-Suástegui, L., Treviño, C., Cajas, J.: Thermal nonlinear oscillator in mixed convection. Phys. Rev. E 84, 046310 (2011)CrossRefGoogle Scholar
  39. 39.
    Martínez-Suástegui, L., Treviño, C., Cajas, J.: Steady and oscillatory laminar opposing mixed convection in a vertical channel of finite length subjected to symmetrical isothermal discrete heat sources. Phys. Fluids 27, 063604 (2015)CrossRefGoogle Scholar
  40. 40.
    Martínez-Suástegui, L., Treviño, C.: Transient laminar opposing mixed convection in a differentially and asymmetrically heated vertical channel of finite length. Int. J. Heat Mass Transf. 51, 5991–6005 (2008)CrossRefMATHGoogle Scholar
  41. 41.
    Meneghini, J., Saltara, F., Siqueira Jr., C., Ferrari, J.: Numerical simulation of flow interference between two circular cylinders in tandem and side-by-side arrangements. J. Fluid. Struct. 15, 327–350 (2001)CrossRefGoogle Scholar
  42. 42.
    Mettu, S., Verma, N., Chhabra, R.: Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel. Heat Mass Transf. 42, 1037–1048 (2006)CrossRefGoogle Scholar
  43. 43.
    Mittal, R., Dong, H., Bozhurttas, M., Najjar, F., Vargas, A., von Leobbecbe, A.: A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys 227, 4825–4852 (2008)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Mussa, A., Asinari, P., Luo, L.S.: Lattice Boltzmann simulations of 3D laminar flows past two tandem cylinders. J. Comput. Phys. 228, 983–999 (2009)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Nejat, A., Abdollahi, V., Vahidkhah, K.: Lattice Boltzmann simulation of non-Newtonian flows past confined cylinders. J. Non-Newton. Fluid. Mech. 166, 689–697 (2011)CrossRefMATHGoogle Scholar
  47. 47.
    Nejat, A., Mirzakhalili, E., Aliakbari, A., Niasar, M., Vahidkhah, K.: Non-Newtonian power-law fluid flow and heat transfer computation across a pair of confined elliptical cylinders in the line array. J. Non-Newton. Fluid. Mech. 171, 67–82 (2012)CrossRefGoogle Scholar
  48. 48.
    Patankar, S.: Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Coorporation, NY (1980)MATHGoogle Scholar
  49. 49.
    Sarkar, S., Dalal, A., Biswas, G.: Mixed convective heat transfer from two identical square cylinders in cross flow at \(Re = 100\). Int. J. Heat Mass Transf. 53, 2628–2642 (2010)CrossRefMATHGoogle Scholar
  50. 50.
    Sharman, B., Lien, F.S., Davidson, L., Norberg, C.: Numerical predictions of low reynolds number flows over two tandem circular cylinders. Int. J. Numer. Methods Fluids 47, 423–447 (2004)CrossRefMATHGoogle Scholar
  51. 51.
    Slaouti, A., Stansby, P.: Flow around two circular cylinders by the random- vortex method. J. Fluid Struct. 6, 641–670 (1992)CrossRefGoogle Scholar
  52. 52.
    Sumner, D.: Two circular cylinders in cross-flow: a review. J. Fluid. Struct. 26, 849–899 (2010)CrossRefGoogle Scholar
  53. 53.
    Tezduyar, T., Glowinski, R., Liou, J.: Petrov–Galerkin methods on multiply connected domains for the vorticity-stream function formulation of the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 8, 1269–1290 (1988)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Tezduyar, T., Liou, J., Ganjoo, D., Behr, M.: Solution techniques for the vorticity-streamfunction formulation of the two-dimensional unsteady incompressible flows. Int. J. Numer. Methods Fluids 11, 515–539 (1990)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Tezduyar, T., Liou, J.: On the downstream boundary conditions for the vorticity-stream function formulation for two-dimensional incompressible flows. Comput. Methods Appl. Mech. 85, 207–217 (1991)CrossRefMATHGoogle Scholar
  56. 56.
    Thom, A.: The flow past circular cylinders at low speeds. Proc. R. Soc. A 141(845), 651–669 (1933)CrossRefMATHGoogle Scholar
  57. 57.
    Xu, G., Zhou, Y.: Strouhal numbers in the wake of two inline cylinders. Exp. Fluids 37, 248–256 (2004)CrossRefGoogle Scholar
  58. 58.
    Zdravkovich, M.: Review of flow interference between two circular cylinders in various arrangements. J. Fluids Eng. 99, 618–633 (1977)CrossRefGoogle Scholar
  59. 59.
    Zdravkovich, M.: Flow induced oscillations of two interfering circular cylinders. J. Sound Vib. 101, 511–521 (1985)CrossRefGoogle Scholar
  60. 60.
    Zdravkovich, M.: The effects of interference between circular cylinders in cross flow. J. Fluid. Struct. 1, 239–261 (1987)CrossRefGoogle Scholar
  61. 61.
    Zhang, P., Wang, J., Huang, L.: Numerical simulation of flow around cylinder with an upstream rod in tandem at low Reynolds numbers. Appl. Ocean Res. 28, 183–192 (2006)CrossRefGoogle Scholar
  62. 62.
    Zhang, Y., Chen, Z.: Effect of gap between layers on the heat transfer performance of alinged tube banks. Heat Trasnf. Eng. 13, 33–41 (1992)CrossRefGoogle Scholar
  63. 63.
    Zhou, Y., Yiu, M.: Flow structure, momentum and heat transport in a two- tandem-cylinder wake. Exp. Fluids 548, 17–48 (2006)Google Scholar
  64. 64.
    Zukauskas, A.: Heat transfer from tubes in cross flow. Adv. Heat Transf. 8, 93–160 (1972)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Erick Salcedo
    • 1
  • Juan C. Cajas
    • 2
  • César Treviño
    • 3
  • Lorenzo Martínez-Suástegui
    • 4
  1. 1.Departamento de Termofluidos, Facultad de IngenieríaUNAM, MéxicoDistrito Federal, MexicoMexico
  2. 2.Barcelona Supercomputing Center (BCS-CNS)BarcelonaSpain
  3. 3.UMDI, Facultad de CienciasUniversidad Nacional Autónoma de MéxicoSisalMexico
  4. 4.ESIME AzcapotzalcoInstituto Politécnico NacionalMéxico, Ciudad de MéxicoMexico

Personalised recommendations