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

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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.

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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

u, v :

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, \(V = v/u_{0}\)

x, y :

Cartesian rectangular coordinates

X :

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

Y :

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

\(\alpha \) :

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

0:

Ambient or reference

1, 2:

Refers to the upstream and downstream cylinder, respectively

w :

At the surface of the cylinders

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Authors and Affiliations

  1. Departamento de Termofluidos, Facultad de Ingeniería, UNAM, México, 04510, Distrito Federal, Mexico, Mexico

    Erick Salcedo

  2. Barcelona Supercomputing Center (BCS-CNS), Edificio NEXUS II, C/Jordi Girona 29, 08034, Barcelona, Spain

    Juan C. Cajas

  3. UMDI, Facultad de Ciencias, Universidad Nacional Autónoma de México, Sisal, Yucatán, Mexico

    César Treviño

  4. ESIME Azcapotzalco, Instituto Politécnico Nacional, Avenida de las Granjas No. 682, Colonia Santa Catarina, Delegación Azcapotzalco, 02250, México, Ciudad de México, Mexico

    Lorenzo Martínez-Suástegui

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  1. Erick Salcedo
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  2. Juan C. Cajas
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  3. César Treviño
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  4. Lorenzo Martínez-Suástegui
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Correspondence to Lorenzo Martínez-Suástegui.

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Communicated by Rupert Klein.

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Salcedo, E., Cajas, J.C., Treviño, C. et al. Numerical investigation of mixed convection heat transfer from two isothermal circular cylinders in tandem arrangement: buoyancy, spacing ratio, and confinement effects. Theor. Comput. Fluid Dyn. 31, 159–187 (2017). https://doi.org/10.1007/s00162-016-0411-z

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