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


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.


Mixed convection Tandem cylinders Blockage ratio Interference effects Bimodal vortex shedding 



Blockage ratio, D/H


Cylinder diameter (characteristic length)


Vortex shedding frequency (Hz)


Gravity acceleration


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


Local heat transfer coefficient


Width of computational domain


Thermal conductivity of fluid


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


Wake closure length of the upstream cylinder


Wake closure length of the downstream cylinder


Length of computational domain


Normal direction


Local Nusselt number (see Eq. 8)


Average Nusselt number (see Eq. 9)


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


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


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


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


Length from the origin to the channel outlet


Standard deviation


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






Fluid temperature at the channel inlet


Temperature at the cylinders’ surface


Fluid velocity at the channel inlet


Longitudinal and transverse velocity components, respectively


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


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


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


Cartesian rectangular coordinates


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


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



Ambient or reference

1, 2

Refers to the upstream and downstream cylinder, respectively


At the surface of the cylinders


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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

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