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Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 69, 2016 - Issue 3
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Original Articles

A comparative study of four low-Reynolds-number k-ε turbulence models for periodic fully developed duct flow and heat transfer

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Pages 234-248 | Received 27 May 2015, Accepted 24 Jul 2015, Published online: 21 Jan 2016
 

ABSTRACT

In this study, streamwise-periodic fully developed turbulent flow and heat transfer in a duct is investigated numerically. The governing equations are solved by using the finite-control-volume method together with nonuniform staggered grids. The velocity and pressure terms of the momentum equations are solved by the SIMPLE algorithm. A cyclic tri-diagonal matrix algorithm (TDMA) is applied in order to increase the convergence rate of the numerical solution. Four versions of the low-Reynolds-number k-ε model are used in the analysis: Launder-Sharma (1974), Lam-Bremhorst (1981), Chien (1982), and Abe-Kondoh-Nagano (1994). The results obtained using the models tested are analyzed comparatively against some experimental results given in the literature. It is discussed that all the models tested failed in the separated region just behind the ribs, where the turbulent stresses are underpredicted. The local Nusselt numbers are overpredicted by all the models considered. However, the Abe-Kondoh-Nagano low-Re k-ε model presents more realistic heat transfer predictions.

Nomenclature

cp=

specific heat

Cµ, C1, C2=

turbulence model constants

D=

dumping function in the k equation

Dh=

hydraulic diameter

E, f1, f2=

dumping functions in the ε equation

fµ=

dumping function in the Prandtl-Kolmogorov relationship, Eq. (14)

Gk=

generation of kinetic energy

h=

rib height

H=

channel height

k=

turbulent kinetic energy

kf=

thermal conductivity of air

L=

length of one periodic module

=

mass flow rate

Nu, Nu(x)=

local Nusselt Number

P=

pressure

=

periodic part of pressure

Pr=

molecular Prandtl number

Prt=

turbulent Prandtl/Schmidt number in Eq. (5)

=

heat flux

=

total heat input

Re=

Reynold number based on channel height, H

Rt, Ry, Rε=

local turbulence Reynolds numbers

T=

temperature

=

periodic part of temperature

Tb=

local bulk temperature

Tw=

local wall temperature

=

turbulent stress tensor

=

turbulent heat flux

uε=

Kolmogorov velocity scale [= (με/ρ)1/4]

uτ=

friction velocity ()

U=

axial mean (time-averaged) velocity

Ui=

mean velocity components in the xj direction (U, V)

Um=

mean velocity in the channel

V=

transverse mean velocity

x, y=

axial and transverse coordinates

xi=

Cartesian coordinates in tensor notation (x, y)

xR=

reattachment length

y+=

dimensionless normal distance from the nearest wall

α=

overrelaxation factor in Eq. (17)

β=

mean channel pressure gradient across a periodic module

γ=

nonperiodic temperature gradient across a periodic module

δij=

Kronecker delta function

ε=

dissipation rate of turbulent kinetic energy

λ=

friction factor

μ=

molecular dynamic viscosity

μt=

turbulent dynamic viscosity

ν=

molecular kinematic viscosity

νt=

turbulent kinematic viscosity

ρ=

air density

σε, σk=

turbulent Prandtl numbers for the k and ε equations

τw=

surface shear stress

Nomenclature

cp=

specific heat

Cµ, C1, C2=

turbulence model constants

D=

dumping function in the k equation

Dh=

hydraulic diameter

E, f1, f2=

dumping functions in the ε equation

fµ=

dumping function in the Prandtl-Kolmogorov relationship, Eq. (14)

Gk=

generation of kinetic energy

h=

rib height

H=

channel height

k=

turbulent kinetic energy

kf=

thermal conductivity of air

L=

length of one periodic module

=

mass flow rate

Nu, Nu(x)=

local Nusselt Number

P=

pressure

=

periodic part of pressure

Pr=

molecular Prandtl number

Prt=

turbulent Prandtl/Schmidt number in Eq. (5)

=

heat flux

=

total heat input

Re=

Reynold number based on channel height, H

Rt, Ry, Rε=

local turbulence Reynolds numbers

T=

temperature

=

periodic part of temperature

Tb=

local bulk temperature

Tw=

local wall temperature

=

turbulent stress tensor

=

turbulent heat flux

uε=

Kolmogorov velocity scale [= (με/ρ)1/4]

uτ=

friction velocity ()

U=

axial mean (time-averaged) velocity

Ui=

mean velocity components in the xj direction (U, V)

Um=

mean velocity in the channel

V=

transverse mean velocity

x, y=

axial and transverse coordinates

xi=

Cartesian coordinates in tensor notation (x, y)

xR=

reattachment length

y+=

dimensionless normal distance from the nearest wall

α=

overrelaxation factor in Eq. (17)

β=

mean channel pressure gradient across a periodic module

γ=

nonperiodic temperature gradient across a periodic module

δij=

Kronecker delta function

ε=

dissipation rate of turbulent kinetic energy

λ=

friction factor

μ=

molecular dynamic viscosity

μt=

turbulent dynamic viscosity

ν=

molecular kinematic viscosity

νt=

turbulent kinematic viscosity

ρ=

air density

σε, σk=

turbulent Prandtl numbers for the k and ε equations

τw=

surface shear stress

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