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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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26

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