![Comparison of the mean drag coefficient and the rms lift coefficient with the results of Tang et al. [18], A * is equal to 0.2 for the first row, 0.4 for the second row, and 0.6 for the third row.](https://www.researchgate.net/publication/359054624/figure/fig2/AS:1130831112347659@1646622555438/Comparison-of-the-mean-drag-coefficient-and-the-rms-lift-coefficient-with-the-results-of_Q320.jpg)
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arXiv:2203.02265v1 [physics.flu-dyn] 4 Mar 2022
Combined active-passive heat transfer enhancement for a partial superhydrophobic
oscillating cylinder
Ali Rezaei Barandagha, Adel Rezaei Barandagha, Jafar Ghazanfariana,∗
aMechanical Engineering Department, Faculty of Engineering, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran.
Abstract
Numerical simulation of convective heat transfer over a stationary and transversely oscillating partial super-hydrophobic
cylinder has been performed using OpenFOAM libraries. Superhydrophobicity of the cylinder surface has been addressed
by means of a partial slip boundary condition. Applying the slip condition to the surface of the stationary cylinder
causes the drag and the rms lift coefficients to reduce by 46 and 75 percent, respectively. It also augments the average
Nusselt number by 55 percent accompanied by a 21 percent increase of the natural shedding frequency. The partially
superhydrophobic cylinder has also been investigated and the effects of slip on different sections of the cylinder surface
have been analyzed. Considering the reduction of force coefficients, it is shown that the application of slip over a 135◦
segment of the surface is an optimum case, resulting in a 47 and 85 percent decrease of the drag and the rms lift
coefficients, respectively. However, the fully superhydrophobic cylinder provides higher heat transfer rates. Regarding
the transversely oscillating cylinder, superhydrophobicity extends the primary synchronization region, and also exhibits
different wake dynamics behavior compared to the no-slip case. The slip over surfaces also causes the average Nusselt
number to become nearly 6 times greater than the no-slip oscillating cylinder at the lock-in condition. Further analysis
based on thermal performance index (TPI) proves that a high value of T P I = 6 can be reached for the superhydrophobic
cylinder.
Keywords: Superhydrophobicity, Partial slip, Oscillating cylinder, Thermal performance index, OpenFOAM
1. Introduction
Flow and heat transfer over a circular cylinder have
been the subject of numerous investigations as a classic
example of flow past bluff bodies. Serving as a benchmark
problem, it plays an important role in understanding the
mechanisms of more complicated applications. Several ex-
perimental and numerical studies have been conducted in
order to measure and calculate the forces exerted on the
body and to help understand complex phenomena such as
flow separation and vortex shedding involved in such flows.
An unsteady heat transfer from the cylinder is also caused
by the unsteady nature of the vortex shedding process.
Previous investigations have shown the local heat trans-
fer rate to become maximum near the front stagnation
point [1, 2].
Several studies have been carried out both numerically
and experimentally regarding an oscillating circular cylin-
der in a flow stream. The instability mechanism that
causes the vortex shedding, namely the Floquete insta-
bility, can be controlled by the oscillation of the cylinder
in certain ranges of frequency and amplitude by means
of a phenomenon known as the lock-in or wake-capture
∗Corresponding author, Tel.: +98(24) 3305 4142.
Email address: j.ghazanfarian@znu.ac.ir (Jafar
Ghazanfarian)
phenomenon. During this range of synchronization the
natural shedding frequency, i.e., the Strouhal number, is
lost and the wake oscillates at a frequency equal to the
frequency of the body motion. In the past decades, many
researchers tried to shed light on this complex fluid-solid
interaction. Bishop and Hassan [3] are among the first
researchers who experimentally studied the flow over a
transversely oscillating circular cylinder. Their findings
showed that the amplitude of the lift and drag forces are
comparable with the response of a simple oscillator un-
der the influence of an applied harmonic force. They also
found that the average drag force and the lift amplitude
increases, while the phase angle between the lift force and
the body motion showed a sudden change when the excita-
tion frequency was close to the natural shedding frequency.
Koopman [4] studied the effect of transverse oscillation
of the cylinder on the wake geometry, reporting that the
cylinder excitation aligned the vortex filaments with the
cylinder axis and the lateral spacing of the vortices de-
creases as the amplitude of oscillation increases. It was
further shown that the synchronization phenomenon only
occurs above a threshold oscillation amplitude, which at-
tains larger values as the forced oscillation frequency de-
viates from the natural shedding frequency. The lock-in
range was also determined at low Reynolds numbers. Grif-
fin [5] further investigated the influence of a variety of ex-
Preprint submitted to Elsevier March 7, 2022
citation conditions on the cylinder wake geometry from
velocity measurements. Through a series of experiments,
it was shown that both the amplitude and the frequency
of forced oscillations at the lock-in condition affect the for-
mation length that was used as a characteristic parameter.
Griffin and Ramberg [6] studied the effect of lateral oscil-
lation of a cylinder on vortex shedding. They matched the
fluid velocities obtained from experimental measurements
with a mathematical model based on the Oseen vortex in
order to evaluate the unknown parameters. They have
also found an inverse relation between the longitudinal
spacing of the vortices and the excitation frequency. In
a study by Bearman and Currie [7], the pressure was mea-
sured at the lock-in state over a transversely oscillating
cylinder at 90 degrees for a broad range of reduced veloc-
ities and oscillation amplitudes. They reported a sudden
phase jump between the cylinder displacement and the
pressure at 90 degrees near the synchronization frequency.
Zdravkovich [8] provided an explanation of the phase jump
observed in previous studies and related it to the timing
of the vortices being shed with respect to the cylinder dis-
placement. Seo et al. [9] examined a two-phase closed ther-
mosyphon as a passive heat transfer device. They investi-
gated dropwise condensation over a hydrophobic surface.
A higher condenser heat transfer coefficient is reported us-
ing a polymer-based hydrophobic coating film
Ongoren and Rockwell [10] implemented the hydrogen
bubble flow visualization technique and showed that for os-
cillation frequencies lower than the natural shedding fre-
quency, vortices were shed when the cylinder was at its
maximum position on the same side. For frequencies of
oscillation above the natural shedding frequency, the vor-
tices were shed on the opposite side of the cylinder’s max-
imum position. In an experimental study by Williamson
and Roshko [11], the effect of oscillation amplitude on the
wake formation has been investigated. They classified the
vortex shedding patterns based on the number of vortices
being shed per oscillation cycle. It was shown that as the
amplitude of oscillation increases, the vortex shedding pat-
tern changes from a pair of vortices on one side (2S) to a
single vortex on the other side (P+S). They also reported
that as the reduced velocity exceeds a certain threshold
value for a given oscillation amplitude, the pattern de-
viates from the 2S mode to a much more complex form
known as the 2P mode, in which two pairs of vortices are
shed during each cycle. Furthermore, they related the sud-
den phase jump to the immediate transition between the
2S and 2P modes observed in previous studies.
Using the PIV and PTV techniques, Gu et al. [12] car-
ried out an experimental research in which the influence of
oscillation frequency on the timing of vortex shedding was
approved. In addition, with the oscillation frequency sur-
passing the natural shedding frequency, two saddle points
were observed in the streamline pattern. In a numerical
study, Hurlbut et al. [13] used the finite-difference method
to simulate the flow over a cylinder oscillating in both
transverse and streamwise directions. Later, the vortex
shedding characteristics were numerically investigated by
Lecointe and Piquet [14] for in-line and lateral oscillations.
Meneghini and Bearman [15] solved the oscillatory flow
over a circular cylinder using a discrete vortex method
at Re = 200. While determining the lock-in boundary,
they extended the oscillation amplitude up to 60 percent
of the cylinder diameter with the frequency of oscillation
varying around the natural shedding frequency. Kumar et
al. [16] conducted pool boiling experiments over hetero-
geneous wettable surfaces. They printed the polymethyl
methacrylate (PMMA) polymer on the plain copper and
the hydrophobic polymer on the plain copper and superhy-
drophilic surface to construct three different cases. They
reached enhancement of 98.5% in heat transfer coefficients.
Anagnostopoulos [17] provided a numerical solution for
the flow over a cylinder that is forced to oscillate trans-
versely using the finite-element technique. The effect of
transverse motion of the cylinder on hydrodynamic forces
and the wake formation along with the determination of
the lock-in boundary were investigated at the Reynolds
number of 106. Tang et al. [18] performed a series of nu-
merical simulations for a cylinder subjected to transverse
oscillation normal to the incoming flow at Re = 200. The
phase difference between the lift coefficient and transverse
displacement, energy transfer between fluid and cylinder
as well as the associated vortex shedding modes were ad-
dressed and factors that can potentially affect the sign of
the phase difference and energy transfer were investigated.
In an experimental research concerning the vibration of
a cylinder normal to an airstream by Sreenivasan and Ra-
machandran [19], insignificant variation of the heat trans-
fer coefficient was reported for a maximum velocity am-
plitude of 0.2. Saxena and Laird [20] examined an oscil-
lating cylinder in an open water channel and obtained a
60 percent increase in heat transfer coefficient. In exper-
imental studies by Leung et al. [21] and Gau et al. [22],
the heat transfer magnitude was seen to be affected by
the oscillation amplitude and frequency. Fu and Tong [23]
investigated the flow and heat transfer characteristics of
a transversely oscillating heated cylinder and also found
the heat transfer rate to be significantly enhanced in the
lock-in regime.
Flow control methods in two categories, namely ac-
tive and passive techniques, have been designed to con-
trol the wake behind bluff bodies such as circular cylin-
der. Kang et al. [24] studied the laminar flow over a ro-
tating cylinder in the fully developed stage involving vor-
tex shedding at the Reynolds numbers varying from 47
to 200. The vortex shedding and the wake flow patterns
are expected to be modified by the rotation of the cylinder,
which may lead to suppressed flow-induced oscillations and
amplified lift force. In a numerical study, Ingham and
Tang [25] examined the flow over a rotating cylinder at
Re < 47 and relatively small non-dimensional rotational
speeds (α < 3). They reported that rotation delays the
boundary layer separation, despite the fact that the vor-
tex shedding does not happen in the wake region. Tang
2
and Ingham [26] considered the steady flow at Re = 50 and
100 for the non-dimensional rotational speeds in the inter-
val of 0 < a < 1. Nobari and Ghazanfarian determined
the flow pattern over a rotating cylinder with transverse
oscillations, where the effects of both rotation and cross
flow oscillation on the drag coefficient, the flow field, the
lock-in phenomenon, and the wake pattern have been ana-
lyzed [27, 28]. Flow around a circular cylinder with rotary
oscillations has also been investigated in various studies
both experimentally, such as Filler et al. [29], and numer-
ically, like Baek and Sung [30]. Soheibi et al. [31] and
Amiraslanpour et al. [32] investigated the effect of slotted
fins and upstream/downstream splitters, respectively on
flow characteristics of an oscillating cylinder.
In comparison to active control methods, passive tech-
niques are generally easier to apply. To name a few, geo-
metric shaping can be used to control the pressure gradi-
ent. Fixed mechanical vortex generators and splitters are
also considered to be beneficial for separation control [33].
Many basic ideas in scientific and industrial applications
and advancements can be traced back to nature and nat-
ural phenomena, one of which is the high water repel-
lency of the Lotus leaf [34]. This tendency has inspired
the design and construction of superhydrophobic surfaces,
which exhibit high water droplet contact angles, gener-
ally exceeding 150 degrees [35]. Such behavior results in
self-cleaning properties, high corrosion resistance and also
drag reduction. Common methods of developing super-
hydrophobic surfaces either involve surface coating [36] or
creating certain micro/nano structures and ridges on the
surface [37, 38]. The Cassie-Baxter and the Wenzel theo-
ries elucidate the relationship between the surface rough-
ness and its wettability. Since the liquid phase completely
passes into the roughness grooves, a water-water interface
is formed in the Wenzel theory [39], while in the Cassie-
Baxter theory [40], air or some other gases get trapped
beneath the liquid, inside the grooves, eventually creating
an air-water interface.
A slip boundary has an immense effect on flow pattern,
and the drag and lift forces. It is usually characterized
by a slip length that is an imaginary distance inside the
body starting from the interface, along which the tangen-
tial velocity drops to zero. The no-slip boundary condi-
tion is assumed to be valid when solving the Navier-Stokes
equations in most continuum studies. However, in partic-
ular cases such as micro and nano-scale problems and hy-
drophobic surfaces, this condition may fail [41]. In their
study of flow through thin micro channels, Joesph and
Tabeling [42] directly measured the apparent slip-length
on hydrophobic surfaces. In a numerical study, Priezjev
et al. [43] investigated the effective slip behavior on sub-
strates under shear flow in micro channels with alternat-
ing no-slip and shear-free boundary conditions using both
continuum and molecular dynamics simulations. You and
Moin [44] numerically investigated the effect of alternat-
ing circumferential bands of the slip and no-slip boundary
conditions on the surface of a circular cylinder, which were
periodically distributed with different arc lengths. The slip
length was 2 percent of the cylinder diameter both in the
streamwise and spanwise directions. They reported that
the drag force and the root-mean-square of the lift force
decrease by as much as 75 percent.
Ou et al. [45] used the placement of directional grooves
or riblets on the surface to manipulate the flow field and
create an effective slip for drag reduction. Lund et al. [46,
47] obtained expressions for an effective slip boundary con-
dition in typical cases and extended them to surfaces with
periodic roughness. Quere [48] and Xue et al. [49] obtained
large slip-lengths up to 400 µm on super hydrophobic sur-
faces. The liquid on such surfaces is mostly in contact with
air trapped in either structured or unstructured crevices
made by the surface treatment. Since the viscosity of air
is small, fluid flow over the air-water interface is almost
shear-free that reduces the overall drag force. Vakarelski
et al. [50] showed that this apparent slip effect can be fur-
ther enhanced by providing a coherent layer of air formed
on the surface. The flow around circular cylinders with
the slip effect being uniformly distributed on the surface
was numerically studied by Legendre et al. [51], and the
results proved that with increasing the slip length, onset
of vortex shedding was delayed and the amount of drag re-
duction was increased for a given Reynolds number. Park
et al. [52] improved the thermal performance of an inclined
tube in a two-phase heat exchanger by surface modifica-
tion techniques. They used the electroplating technique
with hydrogen bubbles to create porous microstructures
as cavities on a boiling surface and hydrophobic thin films
of Teflon. they reported about 107% enhancement in the
boiling heat transfer coefficient.
Slip in unsteady flows has been investigated using the
molecular dynamics simulations by Thalakkottor and Mohseni [53].
It is found that slip can be determined by both the shear
rate and its temporal gradient. Further MD simulations
were carried out by Ambrosia et al. [54] and Sun et al. [55]
to obtain equilibrium states of water droplets on groove/ridge
textured surfaces using various groove widths and ridge
heights. Through a series of experimental investigations on
superhydrophobic cylinders with ridges on their surfaces,
the Strouhal number and the length of the recirculation
region in the wake were shown to be increasing while the
rms lift force decreases. It was also reported that superhy-
drophobicity shifts the onset of vortex shedding towards
higher Reynolds numbers [56, 57].
In a numerical study by Mastrokalos et al. [58], an
increase in the non-dimensional slip-length was shown to
have a stabilizing effect on low-Reynolds number flow past
a circular cylinder. Kim et al. [59] determined how the
flow separation was affected by rough hydrophobic sur-
faces. They also investigated the ensuing changes of the
vortical structures in the cylinder wake. The effect of su-
perhydrophobicity on viscous and form drag forces was in-
vestigated by Huang et al. [60] at different Reynolds num-
bers (up to 180) and slip-lengths. The viscous drag was
found to be dominant at small slip-lengths and Reynolds
3
numbers below 100, while the pressure drag had the main
contribution to the total drag at higher Reynolds numbers
and slip-lengths. Zeinali et al. [61, 62] further investigated
the idea of reducing the drag force and the rms lift force
using superhydrophobic surfaces. Considering high manu-
facturing costs and complexities of superhydrophobic sur-
face production, especially at large industrial scales, they
introduced the Janus surface concept by means of partially
superhydrophobic surfaces. They matched their numerical
data to the experimental results obtained by Daniello et
al. [57] for a superhydrophobic cylinder by implementing
a partial-slip boundary condition in OpenFOAM codes.
The lattice Boltzmann approach can be used to simu-
late the boiling heat transfer performance on hydrophilic-
hydrophobic mixed surfaces [63, 64]. It is found that an
appropriate increase of the contact angle can promote the
bubble nucleation on the bottom side and enhances the
nucleate boiling on the surface. Also. the interaction be-
tween the bubbles nucleated at the corners and the bub-
bles on the tops of pillars van enhance the departure of
the bubbles at corners.
In the present study, the effect of superhydrophobic-
ity will be investigated on flow and heat transfer charac-
teristics of a stationary and transversely oscillating circu-
lar cylinder by imposing a partial-slip boundary condition.
For the case of a stationary cylinder, the force coefficients,
the vortex shedding frequency and the average Nusselt
number will be analyzed as well as the local distributions
of key parameters such as the pressure coefficient, the skin
friction and the Nusselt number. Next, the effects of ap-
plying slip on different sections of the cylinder surface will
be investigated for the stationary cylinder. Considering
the transversely oscillating cylinder, the lock-in boundary
will be determined for the fully superhydrophobic cylinder
and the effect of slip on the mean drag and the rms lift
coefficient will be studied along with the average Nusselt
number for different amplitudes and frequencies of oscil-
lation. Also, the wake structure and the vortex shedding
modes will be compared to that of a no-slip transversely
oscillating cylinder. Finally, using the concept of the ther-
mal performance index (TPI), heat transfer enhancement
will be analyzed more thoroughly.
This paper is organized as follows. Section 2 presents
the details of geometry, governing equations and assump-
tions. The details of numerical technique are discussed in
section 3. Next, the validation of results for various flow
and heat transfer characteristics for both the stationary
and transversely oscillating cylinders will be presented in
section 4. In section 5 a compendious discussion of the ob-
tained results is presented and lastly, section ?? concludes
the paper.
2. Governing equations and geometry
Considering two-dimensional viscous flow and heat trans-
fer over a transversely oscillating circular cylinder, the
governing equations are the continuity, momentum, and
energy equations that can be described in the ALE frame-
work as
∂uj
∂xj
= 0 (1)
∂ui
∂t + (uj−ψj)∂ui
∂xj
=−1
ρ
∂p
∂xi
+∂2ui
∂xj∂xj
(2)
∂T
∂t + (uj−ψj)∂T
∂xj
=α∂2T
∂xj∂xj
(3)
where uiis the velocity component, ψiis the grid velocity
component, νis the kinematic viscosity, and αis the ther-
mal diffusivity. It can be seen that by setting ψito zero,
the equations above are reduced to the Eulerian form, and
when the fluid velocity component, ui, is set equal to the
grid velocity component, ψi, the Lagrangian form of the
equations are obtained.
A third-type mixed (Robin) boundary condition known
as the partial-slip condition has been utilized to repre-
sent superhydrophobicty of the cylinder in OpenFOAM
codes [61]. This boundary condition is as follows
u∗
slip + (1 −β)∂u∗
∂y∗wall
= 0 (4)
where u∗
slip is the relative velocity of the fluid at the wall,
and βis an adjustable coefficient on which a parametric
study has been carried out to find a suitable value to close
the equations.
The important non-dimensional numbers are the Reynolds
number defined based on the free-stream speed and the
cylinder diameter
Re =UD
ν(5)
the Strouhal number
St =fS tD
U(6)
where fSt is the frequency of natural vortex shedding be-
hind the cylinder, the Prandtl number
P r =ν
α(7)
and the Nusselt number for iso-temperature boundary con-
dition
NuT=−D(∂ T /∂n)
Ts−T∞
(8)
where Tsand T∞are the cylinder and free-stream temper-
atures equal to 330 and 300 K, respectively. The Nusselt
number for the iso-heat flux boundary condition is as fol-
lows
NuQ=−1
T∗
s−T∗
∞
(9)
where T∗is the non-dimensional temperature and is equal
to T
D(∂T /∂ n). The temperature gradient has been set to
4
10000 K/m for the case of iso-flux boundary condition.
Another important parameter is the thermal performance
index (TPI) which is defined as
Nu/Nu0
(Cd/Cd0)1/3(10)
where Nu0and Cd0represent the initial states of the Nus-
selt number and the drag coefficient, respectively. This
initial state could refer to the stationary case with respect
to oscillation or the no-slip state with respect to the super-
hydrophobic walls. The motion of the cylinder is defined
by the following equation
y(t) = Asin(2πf t) (11)
where A and fare the amplitude and frequency of oscil-
lation, respectively. The dimensionless forms of the os-
cillation amplitude and frequency are A∗=A/D and
F∗=f/fSt. In this paper the values of A∗are equal to 0.2,
0.4, 0.6, 0.8 and F∗is varied in a way that both the non-
lock and the lock-in cases occur in the simulations. That
is, for the Reynolds number of 200, the non-dimensional
oscillation frequency is taken to be 0.5, 0.8, 1, 1.2, 1.5, 2.
It should be noted that for the case of the superhydropho-
bic cylinder, the non-dimensional oscillation frequency of
0.1 has also been taken into account. In addition, the re-
sults of the present paper have been generated for the flow
of water at Re = 200 and P r = 7.5.
A cylinder with the diameter of D has been placed
in a cross-flow. The two-dimensional computational do-
main consists of an inflow patch which is placed at x =
-20D, the top and the bottom boundaries are located at
y=±20D, and the outflow patch is placed at a distance of
50D from the center of the cylinder. Respective bound-
ary conditions are uniform distributions of velocity and
temperature, zero-gradient condition for pressure at the
inlet; zero fixed-value for pressure, zero-gradient for veloc-
ity and temperature at the outlet; and both the no-slip
and the partial-slip along with the iso-temperature and
iso-flux boundary conditions over the cylinder surface. A
two-dimensional structured mesh has been generated with
higher mesh density near the cylinder in order to accu-
rately resolve the boundary layer behavior close to the
cylinder surface. Details of the computational domain and
the mesh as well as the aforementioned boundary condi-
tions can be seen in Fig. 1.
3. Numerical method
The finite-volume method has been used through the
open source computational fluid dynamics code, i.e., Open-
FOAM. OpenFOAM is a set of C++ libraries and tools
aimed at solving the problems of continuum mechanics,
specially CFD applications, by means of several discretiza-
tion techniques and numerical solvers, as well as various
pre/post-processing utilities. Laminar incompressible flow
Figure 1: Details of the C-type structured computational mesh topol-
ogy and sizes of the domain.
over a circular cylinder with transverse oscillation has been
solved using OpenFOAM codes. To do so, a dynamic-mesh
flow and energy solver, called ThermalPimpleDyMFoam,
previously developed by Ghazanfarian and Taghilou [65]
has been used.
Regarding the discretization of the time derivatives and
the gradient terms, the Euler method and the Gauss-linear
scheme were used, respectively. For the diffusion terms, a
second-order Gauss-linear method was implemented and
a second-order upwind scheme was used to discretize the
convective terms. For the purpose of coupling the pressure
and velocity, the PIMPLE algorithm was used. This algo-
rithm can be thought of as a combination of the PISO and
SIMPLEalgorithms, all being iterative methods. It should
be mentioned that a better stability is gained in PIMPLE
over PISO, especially when dealing with large time-steps.
The under relaxation factors were set to 0.5, 0.7, 1 for
pressure, momentum and energy equations, respectively.
Furthermore, the convergence tolerance for pressure was
10−7and 10−9for other parameters. The Courant number
was also kept less than unity throughout the simulations.
4. Validation and verification
In this section, the obtained results are compared to
the reported data in previous studies to ensure the ac-
curacy of the present simulations. First, the mesh and
time-step size independence tests are carried out, and then
appropriate mesh/time-step sizes for the simulations are
suggested. Next, the acquired results for flow and heat
transfer around the fixed and the oscillating cylinder will
be compared with the available data.
4.1. Mesh/time-step size independence tests
In order to achieve grid-independent results, a set of
simulations have been performed on three different com-
putational meshes at Re = 200. Table 1 summarizes the
details of mesh topologies and corresponding results. As
can be seen, the difference between the obtained values
5
for the drag coefficient, the Strouhal number, and the av-
erage Nusselt number is less than 1 percent for the fine and
finest cases. Also, Fig. 2 shows the variation of pressure
coefficient and Nusselt number on the surface of the cylin-
der for three grid resolutions. It can be observed that the
difference between the fine and finest cases is less than 3
percent for both the pressure coefficient and Nusselt num-
ber distributions. Therefore, the fine grid is sufficient to
perform simulations.
The numerical predictions were also investigated with
respect to three different time-step sizes. The values of
the mean drag coefficient, the Strouhal number, and the
average Nusselt number have been computed as listed in
Tab. 2. The results for the normalized time-step size of
0.005 is very close to the results generated with ∆t∗=
0.0025. Therefore, the time-step size of 0.005 is found to
be sufficient to generate results.
In order to validate the data obtained in the present
study for the case of fixed cylinder, the drag and lift coeffi-
cients (Cdand Cl, respectively), the Strouhal number and
the average value of the Nusselt number have been com-
pared with the previous experimental and numerical re-
sults available in the literature at Re = 200 and P r = 0.71
in Tab. 3. As can be seen, the present computations are in
good agreement with those calculated before, showing an
overall difference of 5 percent. The local variation of the
pressure coefficient and the Nusselt number have been il-
lustrated in Fig. 3(a) and (b), respectively. The difference
between the current calculations and the previous data is
under 15 percent. Note that the local Nusselt number
around the cylinder has been computed at Re = 100 in
order to be comparable with the results obtained in the
previous studies.
Next, the lock-in boundary, the time-history of the
Nusselt number, the mean value of the drag and the rms
lift coefficients have been verified by the previous data
at hand for the case of transversely oscillating cylinder.
The primary synchronization range, i.e., f /fSt ≃1, has
been determined and compared with the map of Leontini
et al. [74] in Fig. 4(a), showing an overall deviation of 3
percent from the previously obtained results. Figure 4(b)
shows the time-history of the Nusselt number for the oscil-
lating cylinder case at two different oscillation conditions.
The present values have been verified by the results of Fu
and Tong [23], indicating a good agreement between the
two computations with a maximum error of 3 percent. The
mean value of the drag coefficient and the rms lift coeffi-
cient have also been compared with the results obtained
by Tang et al. [18] in Fig. 5 for various frequency ratios
and oscillation amplitudes where a total difference of 10
percent is seen between the computed results and the pre-
vious data.
5. Results
In this section the results of applying superhydropho-
bicity on the flow and heat transfer characteristics of the
stationary and transversely oscillating cylinder will be pro-
vided. For the case of a stationary cylinder, mean force
coefficients and heat transfer rates will be analyzed along
with the local distributions of flow and heat transfer pa-
rameters. Also, the application of slip along different seg-
ments of the cylinder surface will be investigated for the
stationary case. Next, the lock-in boundary and average
drag and lift coefficients will be studied for the oscillating
cylinder. Furthermore, the effect of superhydrophobicity
on the distribution of vorticity and vortex shedding modes
will be investigated as well as heat transfer rates and tem-
perature contours. Finally, the relative variations of drag
coefficient and Nusselt number will be analyzed by means
of the Nu/Cdratio and thermal performance indices.
5.1. Stationary cylinder
In order to investigate the influence of the partial-slip
condition on the flow and heat transfer characteristics of
the stationary cylinder, we first examine the effect of vary-
ing the slip coefficient. Note that β= 1 corresponds to
the no-slip condition, hence a decrease in βin the range
0< β < 1 leads to an increased amount of slip. Similar
to the results reported by Legendre et al. [51], there exists
a threshold value for the slip coefficient below which vor-
tex shedding does not occur and the wake remains steady.
This trend can be seen in our data illustrated in Fig. 6(a)
as the lift amplitude decreases with increasing the amount
of slip and drops off to zero at βcr = 0.05. In order to
shed light on such behavior, Fig. 6(b) demonstrates the
vorticity contours for three cases of β= 0.02,0.1,1. It can
be seen that the wake turns to steady state for the case of
β= 0.02, and the superhydrophobic condition causes the
vortices to become more stretched alongside the stream-
wise direction compared to the no-slip cylinder (β= 1).
Figure 7(a) presents variations of the force coefficients
as βattains different values. It is found that the values of
the normalized mean drag and lift amplitude decrease as
βattains lower values, showing that the force coefficients
can be reduced up to 90 percent when the slip coefficient
reaches zero. It should be noted that the values of drag
coefficient have been normalized with respect to both the
reference no-slip and shear-free values, such that C∗
d=
(Cd−Cd(0))/(Cd(1) −Cd(0)), where Cd(0) and Cd(1) are
the values of drag coefficient for the shear-free and no-slip
conditions, equal to 0.131 and 1.355, respectively. Also,
C∗
l=Cl/Cl(1), where Cl(1) is the no-slip lift amplitude,
equal to 0.690. Variation of the average Nusselt number
is also shown in Fig. 7(b) for iso-temperature and iso-flux
boundary conditions. It is obvious that both curves exhibit
the same behavior under different slip conditions. As a
result, it can be seen that the heat transfer rate is enhanced
for about 100 percent with applying increased slip on the
surface. It should be noted that only the iso-temperature
boundary condition has been considered hereinafter.
In order to further examine the behavior of different
flow parameters for the superhydrophobic cylinder, the
6
Table 1: Results of the mesh independence test for the flow around the fixed cylinder at Re = 200 and P r = 0.71.
Grid CdSt Nu No. of cells No. of cells on the cylinder
Coarse 1.419 0.200 7.739 70178 158
Fine 1.355 0.193 7.444 118736 160
Finest 1.358 0.193 7.508 155134 170
θ
θ
Figure 2: Local variation of the time-averaged (a) pressure coefficient, and (b) Nusselt number for three grid resolutions at Re = 200 and
P r = 0.71.
θ
θ
θ
Figure 3: Local variation of the time-averaged (a) pressure coefficient, data obtained from Norberg [72] at Re = 200, and (b) Nusselt number,
data obtained from Eckert [1], Fu and Tong [23] and Patnaik et al. [73] at Re = 100.
7
Figure 4: Comparison of (a) primary synchronization region with the map of Leontini et al. [74], and (b) temporal variation of the Nusselt
number at f/fS t = 1 for two different oscillation amplitudes of A/D = 0.4 (the lower curves) and A/D = 0.8 (the upper curves), data
obtained from Fu and Tong [23].
Table 2: Results of the time-step size independence test for the flow
around the fixed cylinder at Re = 200 and P r = 0.71.
CdSt Nu Normalized time step (∆t∗)
1.350 0.191 7.654 0.01
1.355 0.193 7.444 0.005
1.356 0.193 7.447 0.0025
link between the slip coefficient and the slip length needs
to be pointed out. Figure. 7(c) shows the variation of
the normalized lift amplitude alongside with the results
of Legendre et al. [51], where Kn is the non-dimensional
slip length. The relationship between the slip length and
the slip coefficient could be attained using this figure, such
that the corresponding values of the mentioned parameters
are plotted against each other in Fig. 7(d). This diagram
could be used to associate the value of slip length to a de-
sired slip coefficient. For the purpose of determining the
accuracy of this diagram, K n = 0.2 is selected, where the
corresponding value of the slip coefficient is found to be
β= 0.1. The values of the normalized drag coefficient and
lift amplitude as well as the normalized Strouhal number
(St∗=St/St(1), where St(1) is the Strouhal number for
the no-slip case, i.e. 0.193) are stated in Tab. 4. It can be
seen that the overall difference between the results is less
than 5 percent.
We choose β= 0.1 to further examine the effect of
superhydrophobicity on various parameters of the flow.
Different flow and heat transfer parameters are shown in
Tab. 5 for the no-slip and superhydrophobic cylinders. Re-
sults in the table indicate that the total drag coefficient has
decreased by 46.2 percent. Slip also causes an almost 50
percent decrease in the form drag accompanied by a 40 per-
cent reduction in the friction drag. The amplitude and rms
of the lift coefficient have been pronouncedly suppressed
both by an amount of 75 percent. The Nusselt number is
also 55 percent higher in the case of the superhydrophobic
cylinder. It can also be seen that the ratio of N u/Cdgoes
up significantly by applying the slip condition, showing a
189.59 percent increase. Lastly, the separation angle is
20.51 percent higher for the superhydrophobic case.
The local distribution of form drag and the skin friction
coefficient are also shown in Fig. 8(a) and (b), respectively.
As can be seen, the pressure difference between the front
and rear stagnation points shows a 24 percent decrease for
the superhydrophobic cylinder, which is responsible for the
pressure drag reduction mentioned above. Furthermore,
superhydrophobicity significantly reduces the skin friction
coefficient over most of the cylinder surface, showing a
65 percent decrease of its maximum value. Figure 8(c)
illustrates the local variation of the Nusselt number for
both the no-slip and superhydrophobic cylinders. It is
clear that heat transfer is enhanced near the front and
rear stagnation points. For both cases, the highest value
of the Nusselt number is attained at θ= 0◦, which is the
front stagnation point, and the lowest value lies between
8
Table 3: The mean drag coefficient, amplitude of the lift coefficient, the Strouhal number, and the average Nusselt number for flow over a
stationary circular cylinder at Re = 200 and P r = 0.71.
Literature data CdClSt N u
Persillon & Braza [66] 1.345 0.7 0.204 -
Liu et al. [67] 1.31±0.05 0.69 0.192 -
Qu et al. [68] 1.32±0.01 0.66 0.196 -
Kim & Choi [69] 1.35±0.05 0.7 0.197 -
Churchill & Bernstein [70] - - - 7.227
Bergman et al. [71] - - - 7.453
Present study 1.355 0.693 0.193 7.444
Figure 5: Comparison of the mean drag coefficient and the rms lift coefficient with the results of Tang et al. [18], A∗is equal to 0.2 for the
first row, 0.4 for the second row, and 0.6 for the third row.
9
Figure 6: (a) Variation of the lift coefficient amplitude against the slip coefficient in the steady and vortical regimes, and (b) comparison of
the wake structure for three cases of β= 0.02, 0.1, 1.
Table 4: Results of the corresponding slip length and slip coefficient
values for the flow around the fixed superhydrophobic cylinder.
C∗
dC∗
lSt∗
Legendre et al. [51], K n = 0.2 0.436 0.257 1.228
Present study, β= 0.1 0.486 0.249 1.207
the separation point and the rear stagnation point. There-
fore, it can be observed that applying the slip condition
increases the heat transfer rate throughout most of the
cylinder surface, starting from the front stagnation point
up until where separation occurs. This matter is further
analyzed for the front stagnation point in Fig. 9(a), where
the temperature profile has been plotted along the stream-
wise direction, starting from the surface of the cylinder
into the upstream field. As is depicted, the temperature
gradient is higher for the superhydrophobic cylinder at the
front stagnation point which explains the larger value of
Nu at this position. This trend stems from higher rates
of convection due to slip in this region, as shown by the
velocity vectors in Fig. 9(b) and (c). Spectral analysis of
the lift coefficient is also carried out by means of the fast
Fourier transform and the result is reported in Fig. 8(d).
The figure proves that superhydrophobicity increases the
dimensionless vortex shedding frequency, i.e., the Strouhal
number by almost 21 percent, from 0.193 to 0.233. This in-
creasing trend is in accordance with the results reported in
previous studies [61]. It is seen that the maximum value
of the normalized power density reduces remarkably for
the superhydrophobic cylinder, which shows a 70 percent
decrease. Finally, Fig. 8(e) depicts the variation of the
mean dimensionless slip-velocity along the surface of the
cylinder, which has been normalized using the free-stream
velocity. It is shown that the amount of U∗
slip rises to its
maximum value at θaround 65◦, and after falling down to
zero at the separation point, remains extremely low in the
wake region.
Next, the characteristics of flow and heat transfer over
partially superhydrophobic stationary cylinders are inves-
tigated. To proceed, five different cases have been consid-
ered which are depicted in Fig. 10(a) to (d), representing
the application of slip over the front half, rear half, upper
half, 45◦slip/no-slip sections and 135◦section of the cylin-
der, respectively. The average values of Cd,Clrms,S t,N u
and Nu/Cdare reported in Tab. 6 for the aforementioned
partially superhydrophobic cases. As can be seen, apply-
ing slip to the front and upper halves of the cylinder results
in lower values of Cdand Clrms compared to the rear half
case. Also, the Strouhal number and N u/Cdattain larger
values for the front and upper half cases. Furthermore, for
the case of alternating slip/no-slip sections, i.e. case (d),
the values of flow and heat transfer parameters lie in be-
tween the front, upper, and rear half values. Lastly, it can
be deduced that applying slip throughout a 135◦section
of the cylinder surface leads to higher amounts of drag and
lift reduction along with an increase in heat transfer rate.
However, the value of N u is higher for the case of a fully
superhydrophobic cylinder.
In order to further analyze the previously mentioned
trends, local distribution of the pressure and skin friction
coefficients are displayed in Figs. 11 and 12 for the five
cases of partially superhydrophobic stationary cylinder, re-
10
β
β
β
β
β
Figure 7: Variation of (a) the force coefficients, (b) the Nusselt number against the slip coefficient, (c) the normalized lift amplitude, and
(d) the corresponding values of slip coefficient against the non-dimensional slip length for the case of stationary cylinder, data obtained from
Legendre et al. [51].
Table 5: Values of the form drag coefficient, the viscous drag coefficient, the total drag coefficient, the amplitude and rms of the lift coefficient,
and the Nusselt number obtained for the stationary cylinder regrading the no-slip and the superhydrophobic conditions (β= 0.1).
Cdp Cdf CdClClrms Nu N u/Cdθsep
The no-slip case 1.077 0.278 1.355 0.690 0.487 18.114 13.368 111.640◦
The superhydrophobic case 0.548 0.178 0.726 0.172 0.121 28.106 38.713 134.545◦
Percentage of increase/decrease 49.11 35.97 46.42 75.07 75.15 55.16 189.59 20.51
11
θ
θ
θ
Nu
θ
Figure 8: Local variation of the time-averaged (a) pressure coefficient, (b) skin friction coefficient, (c) Nusselt number, (d) fast Fourier
transformation of the lift coefficient, and (e) the slip-velocity for the no-slip and superhydrophobic stationary cylinders.
12
Figure 9: (a) The temperature profile (kelvin) along the streamwise direction and close-up views of (b) the no-slip and (c) the superhydrophobic
cylinder at the front stagnation point.
13
Figure 10: Schematic of partially superhydrophobic cylinder, case (a)
front half, (b) rear half, (c) upper half, (d) 45◦slip/no-slip sections,
and (e) 135◦section. Note that the solid and dashed lines represent
the superhydrophobic and no-slip segments, respectively.
Table 6: Values of the average drag coefficient, rms of lift coefficient,
Strouhal number and Nusselt number for the five cases of partially
superhydrophobic stationary cylinder. Note that each row corre-
sponds to the cases mentioned in Fig. 10.
CdClrms St Nu N u/Cd
Case (a) 1.023 0.281 0.213 25.754 25.174
Case (b) 1.165 0.335 0.200 19.820 17.012
Case (c) 0.921 0.321 0.212 23.201 25.191
Case (d) 1.105 0.304 0.200 22.420 20.289
Case (e) 0.708 0.102 0.233 26.990 38.121
spectively. As is depicted, the variation of pressure coef-
ficient is nearly the same for the cases of front and rear
half slip. However, the skin friction coefficient is higher
throughout the first quarter of the cylinder surface in case
(b), which results in a larger overall drag coefficient for this
case. Regarding case (c), the pressure coefficient is slightly
lower than the case of a fully superhydrophobic cylinder
up to θ= 225◦. Afterwards, Cpattains higher values in
case (c). Also, skin friction coefficient exhibits the same
behavior as the fully superhydrophobic cylinder in a sim-
ilar range of θ, gaining larger values from 225◦onward.
Alternating slip/no-slip sections, i.e. case (d), leads to the
appearance of several discontinuities and sudden jumps in
the pressure and skin friction coefficient diagrams. These
sudden changes are more pronounced in the first half of
the cylinder, which is θ < 90 and θ > 270. However, shift-
ing between the no-slip and slip conditions does not affect
the behavior of Cpand Cfinside the 90 < θ < 270 range.
Finally, case (d) shows that the application of slip through-
out a 135◦section of the cylinder results in the same trend
of the pressure and skin friction coefficients, although the
overall value of Cfis marginally higher after θ= 135◦
compared to the fully superhydrophobic cylinder.
Figure 13 shows the distribution of Nusselt number
over the cylinder surface for the aforementioned partially
superhydrophobic cases. As can be seen, the overall value
of Nu is lower than that of the fully superhydrophobic
cylinder in all of the no-slip sections, whereas the appli-
cation of slip leads to improved heat transfer rates and
higher Nusselt values.
5.2. Transversely oscillating cylinder
The effect of superhydrophobicity on the characteris-
tics of flow and heat transfer over a transversely oscillat-
ing cylinder is analyzed in this section. First, the lock-in
boundary for the case of a superhydrophobic oscillating
cylinder is examined and compared to the results obtained
for the no-slip case. It is found that the predominant fre-
quency of the lift coefficient is equal to either the natural
shedding frequency or the frequency of the cylinder os-
cillation. Additionally, for higher oscillation amplitudes,
the lift coefficient generally displays multiple frequencies,
one of which synchronizes with the cylinder oscillation fre-
quency.
In order to demonstrate the non-lock and the lock-in
cases, three sets of Lissajous diagrams and their corre-
sponding FFT spectrum have been shown in Fig. 14. It is
well known that the variation of the lift coefficient against
the displacement of the cylinder, i.e., the Lissajous plot,
presents an irregular behavior for the non-lock cases, while
a closed and regular pattern appears for the lock-in condi-
tion. This trend can be seen in Fig. 14(a) when F∗= 0.5
regarding the no-slip cylinder, and cases (b) and (c), re-
garding the superhydrophobic cylinder for the frequency
ratios of 1 and 2, respectively. Also, the FFT spectrum of
the non-lock case (Fig. 14(a)) indicates that the predomi-
nant frequency is equal to the natural shedding frequency,
14
θ
θ
θ
θ
θ
Figure 11: Local variation of the pressure coefficient for the five cases of partially superhydrophobic stationary cylinder, case (a) front half, (b)
rear half, (c) upper half, (d) 45◦slip/no-slip sections, and (e) 135◦section. Note that NS and SH refer to the no-slip and superhydrophobic
segments, respectively.
θ
θ
θ
θ
θ
Figure 12: Local variation of the skin friction coefficient for the five cases of partially superhydrophobic stationary cylinder, case (a) front half,
(b) rear half, (c) upper half, (d) 45◦slip/no-slip sections, and (e) 135◦section. Note that NS and SH refer to the no-slip and superhydrophobic
segments, respectively.
15
θ
θ
θ
θ
θ
Figure 13: Local variation of the Nusselt number for the five cases of partially superhydrophobic stationary cylinder, case (a) front half, (b)
rear half, (c) upper half, (d) 45◦slip/no-slip sections, and (e) 135◦section. Note that NS and SH refer to the no-slip and superhydrophobic
segments, respectively.
i.e., the Strouhal number. This means that the lock-in con-
dition has not happened since the cylinder oscillation fre-
quency is equal to half of the natural shedding frequency.
On the other hand, due to the fact that the cylinder oscil-
lation frequency is the same as the Strouhal number, the
spectrum of data in Fig. 14(b) shows a peak at the fre-
quency ratio of 1, which means that lock-in has occurred.
Figure 14(c) depicts two peaks at the frequency ratios of
around 1 and 2, meaning that both of the natural shedding
oscillation and cylinder oscillation frequencies are present
in the spectrum and therefore, synchronization is lost. It
should be noted that A∗= 0.2 for all three settings.
Fig. 15 presents boundaries of the lock-in occurrence
for various amplitudes and frequencies regarding both the
no-slip and slipped fields. It is seen that the lock-in bound-
ary becomes notably wider on its low-frequency side. Su-
perhydrophobicty also expands the high-frequency bound-
ary in comparison to the no-slip case. Another feature to
note is that the lowest oscillation amplitude for which the
synchronization occurs goes down as a result of slippage.
As mentioned before by Meneghini and Bearman [15], this
lowest amplitude is equal to 0.1 for the no-slip cylinder.
However, this value is found to be around 0.02 for the
superhydrophobic cylinder.
Variation of the mean drag coefficient for various os-
cillation frequencies and amplitudes has been shown in
Fig. 16(a) and (b) for the cases of no-slip and superhy-
drophobic cylinders, respectively. Regarding the superhy-
drophobic case, for A∗= 0.2,0.4,0.6, the value of aver-
age Cdincreases inside the lock-in region, and drops after
reaching a maximum value. The same trend is initially
followed by the no-slip oscillating cylinder, except that for
the case of A∗= 0.8, the mean drag coefficient continues to
grow outside the synchronization range. It should be noted
that the same behavior is not seen for A∗= 0.8 in the case
of the superhydrophobic cylinder. Therefore, it can be
deduced that in the primary synchronization region, the
oscillation amplitude for which the mean drag coefficient
has an ever-increasing trend is shifted to higher values as
a result of superhydrophobicity. Figure 16 also depicts the
variation of Clrms for different oscillation amplitudes and
frequencies in the primary lock-in range. As can be seen
in Fig. 16(c), the rms of lift coefficient decreases inside
the synchronization range, and then attains higher values
at higher oscillation frequencies for the no-slip cylinder.
On the contrary, the superhydrophobic cylinder shows a
completely reversed trend. As Fig. 16(d) illustrates, super
hydrophobicity increases the amount of Clrms inside the
lock-in boundary and outside this range, the rms of the
lift coefficient decreases.
In Figs. 17(a) to (c), the vorticity contours have been
shown for A∗= 0.2 and frequency ratios of 0.5, 1, and 0.2,
respectively. As can be seen, the vortices become more
stretched for the case of a superhydrophobic cylinder at
F∗= 0.5 and are placed at a far more distance in the
wake with respect to each other. At F∗= 1, the slip causes
16
Figure 14: The Lissajous diagrams and corresponding FFT spectrum of (a) the no-slip cylinder at F∗= 0.5, (b) and (c) the superhydrophobic
cylinder at F∗= 1 and 2, respectively. A∗= 0.2 in all cases.
17
Figure 15: Comparison of the lock-in boundaries for the cases of the
no-slip and superhydrophobic cylinder.
the von K´arm´an vortex street to be more compressed and
by increasing the frequency ratio up to 2, the vortices in
the wake begin to part from each other again. Also, the
shape of the vortices close to the cylinder surface differs
at F∗= 2, and depicts a higher level of attachment to the
cylinder surface. Furthermore, the local distribution of
vorticity on the surface of the cylinder has been shown in
Fig. 17 for the same set of oscillation parameters. It is clear
that superhydrophobicity suppresses the vorticity over the
cylinder surface and the vorticity production occurs in the
region θ= 180◦−360◦, while this range is around 120◦−
240◦for the no-slip cylinder.
In accordance with the results of Leontini et al. [74],
the wake structure depends on the oscillation amplitude
when the frequency of oscillation is around 1, resulting in
appearance of the P+S vortex shedding mode at oscilla-
tion amplitudes of higher than 0.7. Figure 18(a) and (b)
demonstrates the wake structure and the vortex shedding
modes for the amplitudes of 0.2, 0.4, 0.6, 0.8 at F∗= 1 for
the cases of no-slip and superhydrophobic cylinder, respec-
tively. The results of the no-slip case are in total agreement
with those of Leontini et al. [74]. For the superhydropho-
bic cylinder, the transformation of the 2S vortex shedding
mode to the P+S mode does not happen by crossing from
A∗= 0.6 to 0.8. However, the vortex streets are formed
near the top and the bottom sides of the cylinder moder-
ately earlier in comparison to the no-slip case.
Variation of the average Nusselt number for the cases
of the no-slip and superhydrophobic cylinder has been
demonstrated in Fig. 19(a) and (b), respectively. It is
found that for the no-slip oscillating cylinder, the mean
Nusselt number increases inside the lock-in range and be-
comes larger with increasing the oscillation amplitude. On
the other hand, the superhydrophobic cylinder has the
same increasing trend inside the synchronization bound-
ary, but experiences a reduction as the oscillation fre-
quency takes higher values. Also, the values of the mean
Nusselt number are significantly higher than the results
for the no-slip cylinder. For instance, the highest value of
Nu for the superhydrophobic cylinder is around 6.5 times
greater than its counterpart for the no-slip case.
Local distribution of the Nusselt number over the sur-
face of the cylinder has been shown in Fig. 20(a), (b), (c),
at A∗= 0.2 and F∗= 0.5,1,2, respectively. The most
significant fact we can detect here is that for the case of
the no-slip cylinder, the Nusselt number decreases from a
maximum magnitude at the front stagnation point, and
attains its lowest value at around θ= 120◦. Then, after
passing through a set of local maxima and minima in the
separated region, it rises again to the same highest value
near the rear stagnation point. However, regarding the su-
perhydrophobic case, a completely different trend can be
exhibited. Most importantly, the highest value of the Nus-
selt number no longer appears close to the front stagna-
tion point. It is observed that the slip condition causes the
Nusselt number to rise to its peak value before dropping to
almost zero in the separated region. Fig. 20 also illustrates
the temperature contours for the same set of parameters
mentioned above. It can be seen that for the case of a
superhydrophobic cylinder, due to the increased amount
of the Nusselt number, a higher rate of heat is transferred
into the flow wake. Furthermore, it is observed that at the
region of θ= 180◦−300◦, the temperature gradient is al-
most zero and the uniform distribution of the temperature
leads to the low values of N u mentioned before.
Figure 21(a) and (b) show the variation of N u/Cdfor
the cases of the no-slip and superhydrophobic oscillating
cylinders, respectively. The value of Nu/Cdfollows an
overall decrease for the case of the no-slip cylinder, reach-
ing a minimum at the lock-in state, i.e., f /fSt = 1. This
trend indicates that heat transfer enhancement can not be
merely attributed to the variation of the Nusselt number,
and the N u/Cdratio provides a more precise analysis of
heat transfer augmentation with respect to the changes
of drag coefficient. On the other hand, as Fig. 21(b) de-
picts, superhydrophobicity alters the trend of N u/Cdin
a major way. Before the lock-in phenomenon occurs, the
value of the Nusselt number divided by the drag coefficient
is rising to a maximum, which is achieved at the lock-in
state, and as the oscillation frequency takes higher values,
Nu/Cddecreases. Consequently, since applying the slip
to the surface of the cylinder enhances the heat transfer
rate while reducing the drag coefficient, it can be deduced
that superhydrophobicity provides a better heat transfer
performance for a wide range of oscillation amplitudes and
frequencies.
The combined analysis of flow and thermal fields can be
further extended by computing the thermal performance
18
Figure 16: Comparison of Cdand Clrms for the cases of no-slip cylinder (left column) and superhydrophobic cylinder (right column).
19
θ
θ
θ
Figure 17: Local variation and the contours of vorticity at the oscillation amplitude of 0.2, (a) F∗= 0.5, (b) F∗= 1, and (c) F∗= 2. In all
frames, the cylinder is at its extreme upper position.
20
Figure 18: Comparison of the wake structure and vortex shedding modes for the cases of (a) the no-slip and (b) the superhydrophobic
oscillating cylinder at F∗= 1 and A∗= 0.2,0.4,0.6,0.8. In all frames, the cylinder is at its extreme upper position.
21
Figure 19: Comparison of the average Nusselt number for the cases of (a) the no-slip and (b) the superhydrophobic cylinder.
θ
θ
θ
Figure 20: Local variation of the Nusselt number and temperature contours for the oscillation amplitude of 0.2, (a) F∗= 0.5, (b) F∗= 1,
and (c) F∗= 2. In all frames, the cylinder is at its extreme upper position.
22
Figure 21: Comparison of the ratio of the Nusselt number and the drag coefficient for the cases of (a) the no-slip and (b) the superhydrophobic
oscillating cylinders, and the thermal performance indices for the cases of (c) the no-slip, (d) and (e) the superhydrophobic oscillating cylinders.
Regarding the TPI curves, note that the reference states are according to those of the stationary cylinder for (c) and (d), and the oscillating
cylinder for (e).
23
index (TPI) for our cases. The variation of TPI is shown
in Fig. 21(c) and (d) for the no-slip and superhydrophobic
cylinder, respectively. As equation (10) suggests, a ref-
erence state is needed to be defined in order to calculate
TPI for each case. In Fig. 21(c) and (d), the values of Nu
and Cdfor the stationary cylinder have been considered as
initial states. Thus, Fig. 21(c) and (d) represent the effect
of oscillation on thermal performance of the cylinder for
both cases of the no-slip and slipped cylinders. For the
no-slip surfaces, the decreasing trend suggests that the in-
crease of the Nusselt number for the oscillating cylinder
can not cope with the higher values of the drag coeffi-
cient at the lock-in state. Therefore, TPI falls down to a
minimum at the lock-in region. This trend supports the
fact that the enhancement of heat transfer can not only
be decided by an increase in the Nusselt number and the
effect of the drag coefficient needs to be considered as well
in applications with constant pumping power. However,
as Fig. 21(d) illustrates, the thermal performance index
experiences an increase up to a maximum at the lock-in
region. This feature is completely in contrast with that
of the no-slip cylinder, showing a reducing trend. As a
result, superhydrophobicity can enhance the thermal per-
formance of the oscillating case by about 5 times due to
the fact that it increases the Nusselt number and simulta-
neously reduces the drag coefficient. The effect of super-
hydrophobicity on TPI has also been investigated for the
oscillating cylinder in Fig. 21(e). It should be noted that
the reference states are defined as the value of the Nusselt
number and the drag coefficient of the oscillating no-slip
cylinder in this case. Similar to Fig. 21(d), the thermal
performance index attains a maximum at the lock-in con-
dition, and exhibits ten times higher values in comparison
to the data in Fig. 21(c). This means that for the case
of an oscillating cylinder, application of slip on the sur-
face highly enhances the heat transfer from the cylinder
compared to the no-slip state.
6. Conclusions
Two-dimensional laminar flow and heat transfer over
superhydrophobic stationary and transversely oscillating
cylinder is numerically studied. After the validation of re-
sults, the effect of slip on the mean flow and heat transfer
characteristics of the stationary cylinder has been studied
by means of analyzing the local distributions of various
flow and heat transfer parameters. It is shown that in-
creasing the amount of slip causes the vortex shedding in
the wake to vanish at a specific threshold value. The mean
drag coefficient, the amplitude of the lift coefficient and the
magnitude of the rms lift coefficient reduce as a result of
increased slip. The average Nusselt number is also shown
to attain higher values for the case of a superhydrophobic
cylinder for both the iso-temperature and iso-flux bound-
ary conditions.
After choosing a fixed value for the slip coefficient, it is
shown that superhydrophobicity reduces the total drag co-
efficient by 46.2 % and increases the Nusselt number by 55
%. Analysis of the local distributions of the pressure and
skin friction coefficients resulted in the reduction of form
and friction drag. Furthermore, the FFT spectrum of the
lift coefficient reveals that the natural shedding frequency
for the superhydrophobic cylinder is about 21 % higher
than that of the no-slip case. The effects of applying slip
over different sections of the cylinder surface are also stud-
ied and the results show that the front and upper halves of
the cylinder are relatively more effective in the reduction
of force coefficients and heat transfer augmentation, with
135◦case being the optimum, resulting in a 47 and 85 %
decrease of the drag and lift amplitude coefficients. How-
ever, the fully superhydrophobic cylinder provides higher
values of Nu, being around 5 % higher than that of the
135◦case.
Regarding the transversely oscillating superhydropho-
bic cylinder, it is demonstrated that the boundary for the
primary synchronization region is expanded on both the
lower and higher frequency limits of the map. Further-
more, the average drag coefficient follows the same trend
as the no-slip case for different amplitudes and frequencies
of oscillation, except the point that the oscillation ampli-
tude for which the mean drag continuously elevates with
respect to the frequency ratio is shifted towards higher val-
ues. On the contrary, the trend of the rms lift coefficient
for the superhydrophobic cylinder is completely different
from the no-slip case, such that after rising to its max-
imum at the lock-in state, it goes down to lower values
with increasing frequency ratio.
The wake structure is then analyzed and compared to
the no-slip case, showing that the formation of the vor-
tices on the cylinder surface and their diffusion into the
wake has been significantly altered as a result of super-
hydrophobicity. Also, for the oscillation amplitudes and
frequencies considered in this paper, the vortex shedding
mode of P+S does not occur for the superhydrophobic
cylinder. The investigation of the Nusselt number and its
local distribution over the surface of the cylinder suggested
that superhydrophobicity increases the heat transfer rate
from the oscillating cylinder compared to the no-slip case.
Further, the analysis of the ratio of the Nusselt number
over the drag coefficient provides more insight on the rela-
tive importance of heat transfer enhancement with respect
to drag reduction. In order to study this matter more
thoroughly, an important variable called the thermal per-
formance index (TPI) has been utilized and the effects of
oscillation and slip have been analyzed on the variation
of this parameter. For the case of the no-slip cylinder,
it is shown that oscillation causes TPI to drop off to a
minimum at the lock-in condition, whereas the superhy-
drophobic oscillating cylinder attains a peak value for TPI
at this state. This means that oscillation has an enhanc-
ing effect on thermal performance of the superhydrophobic
cylinder. The same feature was observed for the thermal
performance of an oscillating superhydrophobic cylinder
with respect to the no-slip oscillating case.
24
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