Template for Electronic Submission to ACS Journals
© Applied Science Innovations Pvt. Ltd., India Carbon – Sci.
Tech. 8/3(2016)153-164
RESEARCH ARTICLE Received: 10/03/2016, Accepted: 25/06/2016
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A Computational study on k-kl-ω model for flow field and heat
transfer on a cylinder in cross-flow
S. K. Dhiman*, J. K. Prasad
Birla Institute of Technology, Mesra, Ranchi-835215, India.
Abstract: In the present computational investigation, the
characteristics of heat transfer from cylinders in cross flow of
air has been reported, using the commercial CFD software, FLUENT.
Unsteady, constant heat flux condition has been applied at the
surface and the turbulent flows were simulated using k-kl-ω
turbulence model. Comparing for single cylinder, with k-ω
(standard), k-ω (SST) and SA models, the k-kl-ω model has been
found best suitable models for simultaneous computation of flow
field and heat transfer parameters. The results were also compared
with experiments performed as well as from reported literatures for
Reynolds number range 1.1x104≤Re≤4.1x104 to prove the capabilities
of k-kl-ω model. High aspect ratio (18.75) and very low blockage
ratio (0.053), has been used to conduct experiments. An analysis of
data revealed that the average deviation from the experimental as
well as reported values is less than 1%.
Keywords: Cylinder; k-kl-ω model; Nusselt number; pressure
coefficient; flow structure
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1 Introduction: Flow and heat transfer past a circular cylinder
is a splendidly investigated benchmark problem for which numerous
literatures exists [1-4]. When flow takes place past such a smooth
body, the vortex-shedding phenomenon occurs over a wide range of
Reynolds numbers. Thus, instabilities behind the cylinder make the
problem critical due to simultaneous determination of both fluid
flow and heat transfer parameters as well as correct flow structure
around the cylinder. Typical examples include heat exchangers like
waste heat recovery recuperators, radiators & condensers,
chemical and food industry processes, reactors, etc. Several CFD
simulations were carried out for numerical investigation to
establish or propose turbulence models viz. k-ε standard, k-ε
Realizable, k-ω (standard) and k-ω SST [5-8].
In several cases, the simulations typically related to heat
transfer, encounter problems of fluid structure interaction. As a
cure, adopting the non-boundary conforming methods based on
Cartesian grid, Peskin [9] introduced an immersed-boundary method,
to address the problems of interaction of complex fluid-structure.
Although this method has been adopted frequently in solving fluid
flow problems, however, computing heat transfer problems with this
method are not as extensive. Kim and Choi [10] extended immersed
boundary method towards computation of heat transfer problems.
Zhang et al. [11] take on the distribution function to enforce the
momentum and the energy along the immersed boundary for the
flow-field and heat transfer simulations past a circular cylinder.
The boundary conditions were isothermal Dirichlet-type and
iso-heat-flux Neumann-type. In addition, investigation of
improvement in heat transfer oscillating the cylinder was reported.
Liao et al. [12] adopted, stationary and moving complex geometry,
to simulate forced and free convection domains using an immersed
method. The method was based on the direct momentum and energy,
imposing on a Cartesian grid, and addressed the issues such as the
implementation of the isothermal and iso-flux boundary
conditions.
Since decades, a constant development in inverse analysis has
now become a valuable alternative due to difficulties in direct
measurements or expensive measuring processes. However, reported
inverse method’s applications in convective heat transfer problems
are much less than those in heat conduction problems. Chen et
al.[13-16], Taler [17], Olsson [18], Benmachiche et al.[19] and
Chen [20] have developed inverse methods for the predictions of
heat transfer at the surface of cylinders. These predictions were
based on iterations and simulations.
In the present investigation 2D CFD simulation of heat transfer
from a circular cylinder under constant heat flux (CHF) condition
in an incompressible transverse flow of air have been investigated
using the commercial CFD software, FLUENT, for the estimation of
fluid flow and heat transfer parameters. The turbulent flows were
simulated using k-kl-ω turbulence model [21]. The capabilities of
this turbulence model to compute lift, drag and pressure
coefficients, Strauhal number and Nusselt number have been verified
for Reynolds number based on diameter ranges from 1.1x104 to
4.1x104.
2. Methodology:
The computational scheme consists of a 2-D circular cylinder
having diameter D, which was maintained at constant heat flux
condition. The temperature difference between the air and the
surface of the cylinder was kept small (10oC), hence the variation
of physical properties, particularly density and viscosity of air,
with temperature could be neglected
The computational domain was O-grid type with cylinder placed
centrally as shown in Figure 1, which was prepared in GAMBIT.
Structured grid was created with a fine mesh very close to the
cylinder surface and coarse grid away from the surface. Grid was
refined till the variation in Pressure coefficient (Cp) and Nu was
less than 0.5%. Finally, after grid independence test the nearest
grid point from the surface was taken at a distance of 0.0128D,
while the outer boundary was at a distance of 60D from the center.
The computational domain was divided into 76000 quadrilateral
cells.
Figure 1: Discretization of Computational domain and a close-up
view
The governing equations of unsteady incompressible flow past a
heated circular cylinder are the typical continuity, N-S and Energy
equations
Continuity:
.
x-momentum:
y-momentum:
Energy Equation:
The boundary conditions are:
Inlet: Air enters the domain along x direction with velocity,
and. Outflow: Zero diffusion flux has been implemented at outflow
boundary i.e. condition of outflow boundary has been extrapolated
from within the domain and has no influence on the upstream flow.
Surface of the cylinder: No slip condition has been applied at the
surface of cylinder and was at constant heat flux condition. .
SIMPLEC method for the pressure linked equations has been used to
solve Navier-Stokes and energy equations.
The discretization of convective terms of momentum equation has
been done using second order upwind scheme and that of the
diffusive term by central difference scheme. The k-kl-ω turbulence
model has been used to predict transition of boundary layer
development from laminar to turbulent regime and calculate
transition onset. The k-kl-ω model is a 3-equation eddy-viscosity
type, which includes transport equations for turbulent kinetic
energy (kT), laminar kinetic energy (kl), and the inverse turbulent
time scale (ω) [22]
.
The inclusion of the turbulent and laminar fluctuations on the
mean flow and energy equation via eddy viscosity and total thermal
diffusivity is as follows:
Effective length if defined as here is the turbulent length
scale and is defined as
Small scale energy is defined as:
where and
Large scale energy is given by:
Turbulent fluctuation term is given by: here small scale
turbulent viscosity is
In which and
where,
is damping function,
is production of laminar kinetic energy by Large scale turbulent
productions,
Large scale turbulent viscosity is modeled as
where
is time-scale-based damping function
; ;
Near wall dissipation is given by:
;
R represents average effect of breakdown of streamwise
fluctuations in turbulence during bypass
transition ,
which is the threshold function controls the bypass transition
process:
The breakdown to turbulence due to instabilities is considered
to be a natural transition production term, given by:
The use of ω as the scale-determining variable can lead to a
reduced intermittency effect in the outer region of a turbulent
boundary layer, and consequently an elimination of the wake region
in the velocity profile.
The damping is defined as:
The Total eddy viscosity and eddy diffusivity are given by:
;
The turbulent scalar diffusivity is defined as:
;
Model constants can be referred from [22]
3. Experimental Setup: Experiments have been performed in a
subsonic open-circuit type wind tunnel with test section 0.6m x
0.6m x1.5 m attached with axial flow fan driven by 7.5 HP, 1500
rpm, AC motor with a speed regulator and digital speed indicator in
the control panel. Figure 2 shows a test cylinder, which was of
99.2% pure copper of thickness 3mm and length 35mm. Its outer
diameter (D) was 32mm. It was fixed between two long aluminium
cylinders of 32mm outer diameter and was separated by two Teflon
plugs. A heating cartridge was pasted on one part of both the
plugs, which fits into the copper cylinder. Another part of both
plugs fits into aluminium cylinders. DC power was supplied to heat
the copper cylinders. T-type thermocouples, 24 in numbers, were
placed below the outer surface around each cylinder at strategic
locations at 15o interval to measure the temperature immediately at
times it was exposed to the stable flow of air. An insulated slider
was kept to cover the copper cylinder that suddenly moved away to
expose the heated cylinder just when the airflow stabilizes in the
wind tunnel. Temperatures were recorded using an NI signal
conditioner, NI Data Acquisition system and a Labview program.
Pressures were recorded using Honeywell pressure sensors.
Temperatures measured were used in inverse procedure to calculate
local heat flux. Experiments were performed for Reynolds number
range from 1.1x104 to 1.1x104, under constant heat flux condition
in an incompressible transverse flow of ambient air. High aspect
ratio (H/D = 18.75) and very low blockage ratio (W/D = 0.053) were
used to conduct experiments.
Figure 2: Experimental setup
Present computational study based on k-kl-ω turbulence model for
flow past single cylinder shows good agreement, which is
represented in Table-1, where Cd is Coefficient of drag and St is
Strouhal number
,
where, f is vortex shedding frequency, D is diameter of cylinder
and U∞ is free stream velocity.
Figure 3 shows the comparison of distribution of Cp for k-kl-ω
model with k-ω (standard), k-ω (SST), SA models and experiments
performed in the present study as well as reported by Igarashi [23]
Re=35,000. It is apparent that Cp distribution on the front part of
the cylinder approximately agrees well in all case, but on the
region of separation flow behind the cylinder Cp deviates greatly.
It is depicted from the results that under CHF condition only for
k-kl-ω model agree well with experiments of Igarashi [23] and
present study. A maximum deviation of 1.2% has been observed with
this model.
Figure 3: Comparison of Cp between k-kl-ω, k-ω (standard), k-ω
(SST), SA models and experiments
Table 1: Comparison of Cd and St based on Reynolds number
Re
100
1.1x104
2.1x104
3.1x104
3.5x104
4.1x104
Author
Braza et. al.
Present Study
Relf (1914)
DVL Hiebtone(1919)
UTIA (1955)
Present study
Relf (1914)
DVL Hiebtone (1919)
Present study
Relf (1924)
Present study
Relf (1914)
UTIA (1955)
Present study
Relf (1914)
UTIA (1955)
Present study
Cd
1.3
1.32
1.15
-
-
1.18
1.18
-
1.2
1.34
1.34
1.3
-
1.32
1.3
-
1.34
St
0.16
0.165
-
0.192
0.212
0.208
-
0.202
0.21
0.192
0.186
-
0.185
0.191
-
0.186
0.19
Figure 4 shows the comparison of distribution of Nu for k-kl-ω
model with k-ω (standard), k-ω (SST), SA models and experiments
performed in the present study as well as reported by Tsutsui and
Igarashi [24] at Re=35,000. It is apparent that Nu distribution on
the front part of the cylinder approximately agrees well in all
case. But on the region of separation flow behind the cylinder Nu
deviates greatly in case of k-ω (SST), Buyruk and Murray.
Distribution of Nu for k-kl-ω model follow closely with
experimental results within 1% deviations but at θ=150o it deviates
a little, which might be due to lack of fineness of grid on this
region. It is also depicted from the Figure 4 that minimum Nu is
achieved about 15o ahead from the point of separation.
Figure 4: Comparison of Nu between k-kl-ω, k-ω (standard), k-ω
(SST), SA models and experiments
Figure 5: Cd , St and CL at Re = 100 (left) and 3.1x104
(right)
The boundary layer generates from the front stagnation point and
its thickness grows as the flow progresses over the cylinder. This
boundary layer is laminar and separates when the low velocity air
adjacent to cylinder surface can no longer sustain the adverse
pressure gradient on the rear side and the flow separates forming a
region of reverse flow close to the surface within the region of
separation and the rear stagnation point. Consequently vortex
establishes on each side on the rear-half of the cylinder. Table-1
presents Cd and St for given entire Reynolds number range in which
present study was carried out for aspect ratio AR=18.75 which shows
a good agreement but Zdrakowich [2], page 64, reported that St
differs by variation in AR of the cylinder. Cd , St and CL are also
shown in Figure 5.
Figure 6 shows the wall pressure distribution compared with the
experimental results performed in the present study as well as
numerical and experimental results of reported literatures. The
well-defined pressure coefficient is Cp=(P-P∞)/(0.5ρU∞2), where P
is the local static pressure on the surface and P∞ is free stream
static pressure. The wall pressure obtained by the present method
agrees well with these experimental and numerical data as depicted
from all the plots in Figure 6. The minimum pressure slowly shifts
away, towards front stagnation point with increase in Reynolds
number, which is the point of separation.
Figure-7 shows the distribution of Nusselt number for the
considered range of Reynolds number. The numerical results of Nu
are compared with the experiments performed in the present study on
heated cylinder under CHF condition. Also, the comparison has been
done from reported literatures based of experiments. It is apparent
that overall Nu increases with increase in Re as shown in Figure 8.
A good agreement at front portion of the cylinder can be seen for
all range of Re. A reasonably good agreement is observed on the
rear part of the cylinder; beyond the separation point till the
reattachment of shear layer occurs i.e. upto ≈135o. Beyond this
point, Nu deviates from the experimental distribution and shifts
towards higher side. It is also observed that with increase in Re
this deviation increases. This could be because of the reason that
the far field outflow boundary is placed at 40D from the center of
the cylinder, which should be increased and because of which there
is higher turbulent intensities of flow instability. Hilpert (1933)
also investigated the influence of temperature difference between
the cylinder and air, Tw – T∞, and found that it could be described
by , reported in[2], page533.
Figure 6: Wall Pressure distribution for Re=100 and Re = 1.1x104
to 4.1x104
Figure 7: Nu distribution for Re=100 and Re=1.1x104 to
4.1x104
Figure 8: Overall average Nu for single cylinder for Re=1.1x104,
2.1x104, 3.1x104, 4.1x104 and 6.2x104
Figure 9 shows the validation of flow structure for k-kl-ω model
around a cylinder with that of typically captured in laboratory in
a water table at Aerodynamics laboratory of B. I. T. Mesra Ranchi.
The visualization was taken with aluminium powder and Canon 7D DSLR
camera at Re=3.5x104.
Figure 9: Flow structures (pathlines) at Re= 3.5x104
5. Conclusions: A computational investigation has been carried
out for single cylinder in cross-flow condition for 1.1x104 ≤ Re ≤
4.1x104, to compare the k-kl-ω turbulence model with k-ω
(standard), k-ω (SST), SA models and experiments performed in the
present study as well as reported in literatures. The comparison of
results shows the k-kl-ω model to be an appropriate model for
simultaneous computations of flow field and heat transfer
parameters. The flow field results agree well within 1.2% while
heat transfer results lie within 1% error. However, after θ=150o,
heat transfer results deviates a little, but are in the acceptable
range.
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