# Combined active-passive heat transfer control using slotted fins and
oscillation in turbulent flow: the cases of single cylinder and tube banks

Haleh Soheibi  Zahra Shomali  Jafar Ghazanfarian<EMAIL_ADDRESS>Mechanical Engineering Department, Faculty of Engineering, University of
Zanjan, P.O. Box 45195-313, Zanjan, Iran.  Department of Physics, Faculty of
Basic Sciences, Tarbiat Modares University, Tehran, Iran

###### Abstract

In heat transfer augmenting methods such as radial fins, the heat transfer
enhancement commonly leads to the drag force increment. In the present paper,
slots are inserted over the fins to simultaneously reduce the drag
coefficient. Turbulent convection heat transfer around a cylinder, as well as
oscillating bundle of tubes including the slotted radial fins have been
investigated. The governing equations are solved in two-dimension utilizing
OpenFOAM software based on k-$\omega$ SST closure model. The cases with
various slot location, slot width, the number of the slots, the fin height,
and oscillation frequencies are examined. In all cases, the Reynolds number is
taken to be equal to 5000. Presence of three slots on the fins reduces the
drag coefficient by 23$\%$ and augments the Nusselt number by 76$\%$. In order
to enhance the heat transfer from the bundle of tube, oscillation of tubes and
utilization of the slotted fins are applied. The optimum situation occurs for
a sample with the oscillating third column that shows 3$\%$ increment in heat
transfer relative to that of the fixed case. This is while adding the slotted
fins to the oscillating tube bank increases the heat transfer up to 2.5 times.

###### keywords:

Extended surfaces , Heat transfer enhancement , OpenFOAM , Slotted radial fins
, Turbulent flow,

††journal: Applied Thermal Engineering

##  1 Introduction

Forced convection around cylinder is a momentous engineering issue that is
widely employed in industries, including heat exchangers, heaters,
refrigerators, nuclear reactors, electronic devices, and even suspended bridge
cables. Over the past decades, researchers have been looking for the methods
to improve the heat transfer rate to design cost-effective systems, and also
reduce the energy consumption. In general, the methods for enhancement of heat
transfer are placed into three groups: active methods, passive methods, and
combined methods. Active strategies are the methods in which the heat transfer
improvement occurs by connecting the system to an external energy source.
Passive methods require no external energy source, and usually enhance the
heat transfer coefficient by modifying the geometry to effectively disturb the
flow or additives to change the physical characteristics of the fluid. Surface
oscillation, fluid oscillation, suction and fluid injection are distinguished
active methods. The most frequently used passive methods include utilizing the
expanded surfaces or fins, surface roughening in micro/nanoscale, and the
addition of nano-particles to the fluid. In combined techniques, both active
and passive methods are used simultaneously to enhance the heat transfer.

Increasing the heat transfer coefficient of heat exchangers by the three
mentioned methods, also augments the frictional losses. As a result, there
exists a need to higher pumping power to accelerate the fluid. To optimize the
performance of heat exchangers, both factors should be contemplated. Among the
methods for augmenting the heat transfer coefficient, surface oscillation is
one of the most common ones. The heat exchangers that use this method are
called oscillating heat exchangers [1]. During oscillation, the lock-on
phenomenon occurs, which is due to the synchronization of the excitation
frequency with the vortex shedding natural frequency. Attachment of different
types of fins is also a common passive method that extends the surface of heat
exchanging. It also affects the heat transfer and the friction by modifying
the flow pattern. Many theoretical, numerical, and experimental researches
have been devoted to investigation of effect of parameters such as the
amplitude and frequency of oscillations, the number of fins and their length
on the heat transfer rate and force drag coefficient.

Griffin _et al._ [2] experimentally studied the flow around oscillating
cylinder at low Reynolds number. It has been demonstrated that when the
cylinder oscillates with the frequency equal or near to the vortex shedding
frequency, the length of the vortex formation region is shortened in
comparison to that of the fixed cylinder. Also, the flow structure in the
downstream vortical region is found to be completely influenced by the
amplitude and frequency of excitations. Williamson and Roshko [3] investigated
a cylinder with cross flow oscillations. They have presented different
sketches and also the corresponding modes of vortex shedding. It has been
found that when the frequency of the vortex shedding becomes equal to the
excitation frequency, the lock-in with specified modes takes place. The
different modes of vortex shedding, the lock-on and the non-lock-on regions
are determined in their reported wavelength-amplitude plots.

Cheng _et al_. [4] experimentally studied the forced heat transfer around the
oscillating cylinder in the range of Re$<$4000\. The heat transfer is seen to
be notably augmented by making the cylinder oscillate. They attributed such
enhancement to the lock-on and also the turbulence effects. They also
numerically explored the effect of the lock-on effect on convective heat
transport through the oscillating cylinder at Re$<$300 [5]. They obtained that
the heat transfer performance substantially increases in the lock-on region.
In 1999, Gau _et al._ [6] experimentally investigated thermal transport over a
cylinder oscillating in cross-flow at harmonic, subharmonic, superharmonic,
and nonharmonic frequencies. The dimensionless frequency defined as the
cylinder excitation frequency, $F_{e}$, to the normal frequency of vortex
shedding, $F_{n}$, equals 0.5, 1, 2, 2.5, 3. It is shown that the lock-in
takes place for both $\frac{F_{e}}{F_{N}}$=1, and 3, which itself results in
augmentation of the heat transfer mechanism. The effect of a cylinder placed
in a uniform stream and oscillating about its axis on the forced heat
convection at Re$<$40 have been numerically studied in [7].

The lock-on phenomenon also appears in the case of rotationally oscillating
cylinder. It is reported that the lock-in occurs at excitation frequencies
that are close to the natural frequency. In the lock-on regime, by increasing
the frequency, the heat transfer coefficient enhances; while outside this
region, the oscillation does not much affect the heat transfer. Further, the
transversely oscillating cylinder in a cross flow has been numerically
investigated by Fu and Tong [8]. It has been announced that in the lock-in
regime, the flow and thermal fields approach a periodic state. Also, the
remarkable increase of the heat transfer in the lock-in regime has been
mentioned. In a continuation of the previous works, the influence of
oscillation on heat transfer is inquired experimentally at low Reynolds
numbers [9]. Their results confirmed that at frequencies close to the natural
frequency, a significant augmentation in heat transfer happens. The heat
transfer coefficient is found to be dependent on several factors such as
shortening the length of the vortex formation region, the details of the
vortex roll-up procedure in the cylinder wake, and the transverse cylinder
velocity.

Kumar _et al._[10] introduced new criteria for the lock-in phenomenon by
investigating the flow around an oscillating cylinder at Re=100. They
explained that the lock-in occurs when two following conditions are satisfied.
The first clause is the existence of the most dominant frequency in the power
spectrum of the lift coefficient that matches the frequency of the cylinder
oscillation. The second condition states that if other peaks exist in the
power spectrum, it should be definitely at super-harmonics of the cylinder
oscillation frequency with the value of integral multiples of it. The regime
where only the first of the two conditions is satisfied is referred to as the
transition regime. All other situations correspond to a non-locked state.
Consequently, three flow regimes: the lock-in, transition, and the non-locked
can be recognized.

More, the fluid flow passing through an oscillating tube bank in aligned
arrangements has been numerically explored [1]. To augment the heat transfer,
the first column of the tubes in the lock-in region is vibrated with different
frequencies and amplitudes. Consequently, approximate 25% augmentation of the
overall Nusselt number has been obtained. Furthermore, the effect of
oscillating first column on heat transfer enhancement in low-amplitude motion
is found to be remarkable. It is recommended to oscillate the first column
with low amplitude and the frequency inside the lock-in regime, to reach the
maximum conceivable heat transfer with the lowest construction costs.

By adding nanoparticles into the base fluid as well as oscillating the
cylinder, the heat transfer rate can be enhanced [11]. The study considers low
Reynolds numbers and also takes the thermodynamical properties, temperature-
dependent. It has been acquired that addition of the nanoparticles into the
base fluid, increases the nanofluid viscosity and consequently it lessens the
vortex shedding frequency. Moreover, the maximum 54.82% augmentation of the
drag coefficient and also 9.66% enhancement of the heat transfer coefficient
due to the transverse oscillation of the cylinder is reported. Furthermore,
utilizing both nanofluid and transverse oscillation found to improve the heat
transfer up to 30.3 and 26.8%, respectively, for static and dynamic models.

Besides, the laminar flow around the rotating cylinder with external
longitudinal fins, the situation which one confronts in electrical motors and
generators, is numerically investigated in [12]. Rotation of the cylinder is
found to cause the recirculating flow in the inter-fin space. Further, it has
been shown that the local heat transfer coefficients are large near the front
face of the fin and also close to the fin tips. Also, the cross-flow forced
convection heat transfer has been studied numerically [13, 14].

When Re$<$200, the maximum Nusselt number for a given value of fin height
occurs for an optimum number of fins such that utilizing more of fins make the
Nusselt number decrease. Also, the authors have investigated the effect of
using permeable and solid fins. For a similar cylinder configuration, the
permeable fins found to provide much higher Nusselt number than that of the
traditional solid fins. By increasing the Reynolds number and the fin height,
the Nusselt number increases while it decreases with increment of the number
of fins. More, using one or two permeable fins results in much higher Nusselt
number in comparison to the one obtained for the configuration with eighteen
solid fins. Also, utilizing porous fins notably reduces the weight and cost.

Later, the turbulent natural convection between two horizontal concentric
cylinders with radial fins has been numerically investigated utilizing the
standard k-$\epsilon$ model [15]. For the number of fins between 2 to 12 and
also the Rayleigh number of 106$<$Ra$<$109, the streamlines and the
temperature contours have been presented. It has been obtained that by
increasing the Rayleigh number, the heat transfer rate enhances. Also, the
higher fin heights are shown to have a blocking effect on the flow and reduces
the Nusselt number. Also the laminar free convection of air around a
horizontal cylinder with external longitudinal fins has been numerically
investigated [16].

It has been reported that increasing the Grashof number, the number of fins,
and the fin length enhances the heat transfer rate. For a identical fin
surface, it is found that larger number of fins with smaller length augments
the heat transfer rate. Haldar _et al._ also performed the conjugate numerical
computation [17]. They obtained that the fin thickness has the dominant
influence on the heat transfer. It is established that adding more fins to the
tube gives rise to the enhancement of the heat transfer, if only the fins are
thin. The particle image velocimetry (PIV) has been used to study the wake
region behind a foamed and a finned cylinder in cross-flow [18] with focus on
turbulence. The measurements have been done for a wide interval of the
Reynolds number, 1000$<$Re$<$10000\. Also, the influence of the fin pitch
ratio on wake development and structural loading specifications of the
cylinder with equispaced circular fins have been experimentally accomplished
by the PIV technique [19]. It is obtained that increasing the number of fins
results in augmentation of the mean drag due to raising the skin friction. At
a critical value of the fin pitch, coalescence of the boundary layers between
adjacent fins causes a notable augmentation of the vortex formation region.

Utilizing the OpenFOAM toolbox, the influence of adding straight fins on
thermal and hydraulic properties of the transient heat and fluid flow over a
circular cylinder has been studied [20]. It is obtained that as the fins are
lengthened, increment of the mean drag coefficient takes place, while the
average Nusselt number decreases. Also, it is concluded that the fin
effectiveness is optimum for all Reynolds numbers when the case has four fins.
On the other hand, the effect of adding straight circular fins to the cylinder
has been experimentally investigated in [21]. In this study, the finned
cylinders with different diameter ratios, $D_{f}/D_{r}$, and the same fin
pitch and thickness are investigated. The PIV measurements are performed at
the Reynolds number of Re=2.0$\times$104 correspondent to the sub-critical
flow regime. The notable decrease in the recirculation region followed by an
amplified primary vortex shedding procedure in the wake of the cylinders are
seen. This amplification results in increase of the momentum transport owing
to the periodic Reynolds stresses in the wake of the finned cylinders with a
higher diameter ratio. Moreover, the pressure at the lee side increases due to
the growth of $D_{f}/D_{r}$, which itself causes a reduction in the drag
force.

There are many studies devoted to the reduction of the drag forces and also
suppressing the vortex shedding from the circular cylinders. In 2002, Tsutsui
and Igarashi [22] set a rod upstream of the circular cylinder, to control the
flow around a circular cylinder in air-stream for
1.5$\times$104$<$Re$<$6.2$\times$104. Existence of two flow patterns: with and
without the vortex shedding from the rod has been reported. The flow pattern
varies for different values of the rod diameter and position and also the
Reynolds number. It has been obtained that the total drag comprising the drag
of the rod is 63% reduced in comparison to that of the single cylinder. In the
following, the experimental study of placing a rod upstream of and parallel to
the cylinder to control the flow around the cylinder were performed for
Re=8.2$\times$104 [23]. The resultant force of the cylinder reduces for
spacing between the rod and the cylinder $\alpha$ (the staggered angle of the
rod and cylinder) less than 5∘. When $\alpha$=0∘ and the ratio of the diameter
of the rod relative to the cylinder diameter is equal to 0.5, the maximum drag
reduces to 2.34$\%$ of that of the single cylinder.

The effect of utilizing the dual detached splitter plates, with the same
length as the cylinder diameter, for a circular cylinder on the drag reduction
is numerically studied in [24]. One splitter plate is placed upstream of the
cylinder, while the other splitter is located in the near-wake region. The
stagnation pressure is reduced by the upstream splitter and the downstream one
increases the base pressure by tranquilizing the vortex shedding. The combined
affect results in a notable drag reduction on the cylinder. Also, the
aerodynamic differences between a bare cylinder and a semi-cylindrical roof
have been investigated via wind tunnel experiments by using a circular
cylinder with splitter [25]. It is obtained that a frontal splitter plate
creates a postcritical flow at relatively low Reynolds numbers by the
generated disturbances.

Later in 2017, Klausmann and Ruck [26] experimentally investigated the flow
around cylinder when 3$\times$10${}^{4}<$Re$<$1.4$\times$105. It has been
shown that the thin porous layer on the leeward side of the cylinder, causes
the enhancement of the base pressure on the leeward side of the cylinder that
leads to a decrease in the drag force and dampens the oscillation amplitudes.
The maximum drag reduction of 13.2% has been measured. Furthermore, the drag
suppression for a high-amplitude oscillating cylinder is investigated
utilizing different configurations of the detached-splitter plates [27]. It
has been obtained that the case with the double upstream splitter has the
highest drag reduction. The maximum reduction of the drag force takes place
for the lock-on condition at dimensionless oscillation frequency of
$f^{*}$=1.1. The gap size and the plate length are found to be the least and
the most effective quantities.

As it is clear, there have been many studies dealing with the oscillating
movement of the cylinder and its effects on flow and heat transfer [28, 29,
30]. Also, the effects of the different parameters such as the various types
of fins, including annular and radial ones are investigated. Several methods
for the reduction of the drag coefficient of the cylinder, suchlike utilizing
different kinds of splitters or pipes have been investigated.

In the present study, the effect of existence of slot on a radial fin on
reducing the drag coefficient is investigated. As mentioned in Ghazanfarian
and Bouzari [20], where the fluid flow over a circular cylinder with radial
fin has been investigated, a high drag coefficient appears for a cylinder with
four radial fins. However, to the best of our knowledge, there have been no
study aiming at reduction of the drag coefficient of the cylinder with radial
fin in cross-flow. The present scrutiny will focus on such effect. Here, we
also explore a cylinder with four fins and slots considered in two vertical
fins. The fluid can flow through the slots and affect the drag coefficient and
the heat transfer by changing the flow structure. The effects of different
parameters such as the position of the slot on the radial fin, the width of
the slot, the number of slots, and the height of the slotted fin will be also
investigated. Further, the effect of oscillation of different columns of the
bundle, and also putting up the radial slotted fin on all tubes on enhancement
of the heat transfer in the staggered tube bank will be studied.

This paper is organized as follows. In Sec. 2, the details of geometry, the
governing equations, and assumptions will be presented. The details of the
numerical method are discussed in Sec. 3. Validation of the results for four
different cases of oscillating and fixed cylinders is presented in Sec. 4.
Then, a precise discussion about the obtained results with adequate
engineering criteria is presented in Sec. 5. First, the results for simulating
flow around the cylinder with slotted fin and the effects of geometric
parameters of the slots on flow and the heat transfer are reported. Second,
the slotted fin placed on the oscillating tube bank and impacts of the
oscillation and the fin on heat transfer from the staggered bundle of tubes
are examined. Finally, Sec. 6 presents concluding remarks.

##  2 Geometry and problem description

Schematic geometry of the cylinder with slotted fin has been shown in Fig. 1.
A cylinder with diameter D and four fins, and also with slots on two vertical
fins has been placed in cross-flow. The flow with velocity U0 enters the
channel from the left at temperature T0 and passes the finned cylinder with
surface temperature of Ts.

The distance of the center of cylinder from the channel entrance and the walls
around the channel is taken to be 8 times the cylinder diameter. Also, the
distance of the cylinder from the channel exit equals 20 times of D. Here, the
connected fins are called as number 1 to number 4. The oscillatory motion of
the cylinder in normal-to-flow direction obeys the following relation:

$y=A\ sin\ (2\pi f_{0}t),$ (1)

where A is the amplitude of oscillation and f0 is the excitation frequency. A∗
is the dimensionless amplitude scaled with respect to the diameter. In the
present case, A∗ is taken to be 0.2, 0.4, 0.6. The dimensionless excitation
frequency is the ratio of excitation frequency to the vortex shedding
frequency. In this study, all the oscillations occur in frequencies near to
the vortex shedding frequency and consequently f∗ is close to 1. The
dimensionless parameters are:

$A^{*}=\frac{A}{D};\ f*=\frac{f_{0}}{f_{s}};\ Re=\frac{U_{0}D}{\nu};\
St=\frac{f_{s}D}{U_{0}};\ Nu=\frac{hD}{k}=$ $\frac{\partial T/\partial n\
D}{T_{s}-T_{0}};\ Pr=\frac{\nu}{\alpha};\ \xi=\frac{Q_{Finned}}{Q_{No-Fin}};\
H=\frac{h_{f}}{R};\ r^{*}=\frac{r}{D};$ $e^{*}=\frac{e}{D};\
t^{*}=\frac{t}{D};\ T^{*}=\frac{T-T_{0}}{T_{s}-T_{0}}.$ (2)

The Reynolds number, $Re=\frac{U_{0}D}{v}$, equals 5,000. Here, $v$ is the
kinematic viscosity of the fluid. The dimensionless vortex shedding frequency
is called the Strouhal number, St. In the definition of the Nusselt number,
Nu, $h$ and $k$ are respectively, the heat transfer coefficient and the
thermal conductivity of the fluid. Also, $\partial T/\partial n$ is the
vertical-to-surface temperature gradient. Herein, the Prandtl number, which is
the ratio of $v$ to thermal diffusivity $\alpha$ is considered to be 0.71.

The dimensionless number $\psi$, is introduced as the Fin-Effectiveness to
investigate the performance of the fin. This number is defined as the ratio of
the Heat flux rate from the tube with fin to the heat flux from the tube
without fin (naked tube). The inset of Fig. 1(a), demonstrates the geometrical
parameters of the fin and the slots on it. The height of the fin, hf, is
scaled to the radius of the cylinder. H is the dimensionless length of the fin
scaled to the $D$. In the current study, H is contemplated 0.65, 0.75, 0.85.
$r$ is the distance of the center of slot from the connection point of the fin
to the tube. The location of the slot is described by the dimensionless number
$r^{*}$. The problem is investigated for the slot being in three different
positions of 0.05, 0.15, 0.25. The other important non-dimensional parameter
is the ratio of the width of the slot to the diameter, $e^{*}$, that is taken
to be 0.01, 0.02, 0.06, 0.4. Further, the scaled thickness of the Fin,
$t^{*}$, is set to 0.03.

The thermal resistance of the fin is ignored due to its narrowness and high
thermal conductivity. The temperature at the surface of the tube and the fins
is a constant equal to Ts. The non-dimension temperature, T∗, varies between
zero and one. T${}^{*}=0$ is the non-dimensional temperature of the free-
stream and T${}^{*}=1$ is that of the cylinder surface. The next studied
geometry is the staggered bundle of tubes with slotted fins, see Figs. 1(b)
and (c). As it is seen, the tubes in the tube bundle are placed in four
columns. In [1], where an inline tube bank has been investigated, the maximum
Nu/Cd is found to belongs the case for which the distance between the center
of the tubes is three times the cylinders’ diameter. Herein, as it is seen in
Fig. 1(c), the vertical and horizontal distance between two adjacent cylinders
is taken to be three times the diameter of the tube. Further, the distance of
the inlet boundary from the center of the tubes of first column, the upper
wall from the center first column’s tubes, and also the lower wall from the
tubes of the third column, are the same and equal to the eight times of the
cylinder diameter. The last column is 20D far from the outflow boundary. The
cylinders are named as Ci,j with $i$ and $j$, respectively, denoting the
number of the column and the row. For instance, C32 refers to the tube located
in the third column and the second row. Each column can oscillate
perpendicular to the flow according to Eq. (1).

Figure 1: The studied geometries: (a) a finned tube inside the channel, (b)
bundle of finned tubes inside the channel, (c) two/three-slotted cases.

##  3 Governing equations and closure

The governing equations are the time-averaged version of the mass, momentum,
and energy conservation equations,

$\displaystyle\frac{\partial\bar{u_{i}}}{\partial x_{i}}=0$ (3)
$\frac{\partial\bar{u_{i}}}{\partial t}+\frac{\partial\bar{u_{i}}}{\partial
x_{j}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial
x_{i}}+\nu\frac{\partial^{2}\bar{u_{i}}}{\partial
x^{2}_{j}}-\frac{\partial\bar{u’_{i}}u’_{j}}{\partial x_{j}}.$ (4)
$\left.\begin{aligned} &\frac{\partial{\bar{T}}}{\partial
t}+\frac{\partial{\bar{T}}}{\partial
x_{j}}=\alpha\frac{\partial^{2}\bar{T}}{\partial
x^{2}_{j}}.&\end{aligned}\right.$ (5)

Here, $\rho$, $v$, and $\alpha$ are, respectively, the fluid density, the
fluid kinematic viscosity, and the diffusion coefficient. Moreover,
$\bar{u_{l}}$ is the time-averaged velocity while $u’_{l}$ denotes the
fluctuations of velocity.

The boundary conditions are the no-slip and no-temperature jump, and zero
gradient pressure at the inlet boundary. The K-$\omega$ SST model as a two-
equation model, has been used to simulate the flow. This model, proposed by
Menter in 1993, is a combination of the K-$\omega$ and $K-\epsilon$ models.
The K-$\omega$ model, which is more efficient than the K-$\epsilon$ model in
solving the near wall flow, is more appropriate for the study of boundary-
layer suffering separation undergoing intense adverse pressure gradients. This
turbulent model strongly depends on the boundary conditions of the free-stream
at inlet. On the other side, the K-$\epsilon$ turbulent model, is suitable for
resolving the shear flow like vortex shedding downstream of the bluff bodies
and is less sensitive to the inlet BCs. This model is not adequate for near-
wall flow where the local Reynolds number is low. This model presents more
accurate predictions relative to the other RANS models and, consequently, it
has attracted more attentions in recent years. Here, the RANS discretized
equations of turbulent flow are solved using the finite volume method (FVM)
and the opensource OpenFOAM software. The following relations are the
governing equations in incompressible Newtonian form,

$\displaystyle\frac{D}{Dt}(\rho\omega)$ $\displaystyle=$
$\displaystyle\nabla.(\rho D_{\omega}\nabla\omega)+\frac{\rho\lambda
G}{\nu}-\frac{2}{3}\rho\gamma$ $\displaystyle\omega(\nabla.u)$
$\displaystyle-$
$\displaystyle\rho\beta\omega^{2}-\rho(F_{1}-1)CD_{k\omega}+S_{\omega}$
$\displaystyle\frac{D}{Dt}(\rho k)$ $\displaystyle=$
$\displaystyle\nabla.(\rho D_{k}\nabla k)+\rho G-\frac{2}{3}\rho
k(\nabla.u)-\rho\beta^{*}\omega k+S_{k}$

Here, $k$ is the turbulent kinetic energy and $\omega$ is the rate of
dissipation, which when solved, yields the Reynolds stress tensor. The used
solver is “themalPimpleDyMFoam” that is created by adding the energy equation
to the “pimpleDyMFoam” solver. To discretize the time derivative terms, the
Euler method is used. Also, the advection terms and the diffusion and gradient
terms are, respectively, calculated via the Gauss-linear upwind and the Gauss-
linear schemes.

Overall and close snapshots of the computational mesh are presented in Fig. 2.
As it is seen, near the surface of tubes and fins, the structured boundary
layer mesh is employed while an unstructured mesh is utilized for regions far
from the surface. The size of the first cell on the body is very important
while simulating the turbulent flow. In the near wall region, due to the
intense gradient of parameters under the effect of viscosity, the center of
the first cell of the mesh from the surface should be in a suitable distance
from the wall. So, the non-dimensional number $y^{+}$ is defined as
$\frac{yu_{\tau}}{\nu}$. The friction velocity is a non-dimensional quantity
that is defined by the shear stress on the wall, the representative of the
velocity gradient of the wall, and is explained as
$u_{\tau}=\sqrt{\tau_{w}/\rho}$. The flow in the range 30$<y^{+}<$300 is
completely turbulent and the Reynolds stress tensor overcomes the stresses
resulted by the molecular viscosity. The center of the first cell should be
placed inside the laminar sub-layer, to make the results to be reasonably
accurate. In this study, the mesh is generated in a such way that the $y^{+}$
equals 1. The rate of growth in size of the boundary layer mesh is 1.1 to make
sure that there are appropriate number of cells in the near-wall region.

Figure 2: Unstructured mesh with boundary layer grids, (a) the mesh around the
tube, (b) the mesh around the staggered bundle of tubes, (c) a close up of the
mesh around the staggered tube banks.

##  4 Validation of results

At the first step, the obtained results are compared to the reported data in
previous researches in order to make sure that the present simulations present
acceptable accuracy. Hence, initially, the mesh- and time-step size
independence tests are performed, and then appropriate mesh/time-step sizes
for the simulations are proposed. Secondly, the acquired results for heat
transfer and flow around the fixed and the oscillating cylinders will be
compared with the available data.

###  4.1 Mesh/time-step size independence tests

Simulations have been performed on three different mesh topologies at Re=3900
to achieve the grid-independent results. The details of topologies and
corresponding results are summarized in Tab. 1. When the number of elements of
the computational grid increases from 29,516 elements to 39,516; the obtained
results for the drag coefficient, the Nusselt number, and the Strouhal number,
respectively, change for 0.18$\%$, 0.77$\%$, 0.42$\%$. Consequently, the grid
with total size of 29,516 is adequate to accomplish an authentic simulation.

On the other hand, the numerical results were also investigated using three
different time-step sizes. The Nusselt number, the drag coefficient and the
Strouhal number are calculated then. As it is manifested in Tab. 2, the
results for the normalized time-step of 0.001 is very near to that of 0.0001.
In more details, the percentage change for the drag coefficient and the
Nusselt number is less than 0.15%, and less than 1.3% for the Strouhal number.
Accordingly, the case with 0.001 has been taken to proceed the calculation.

Table 1: The result of mesh independence test for the flow around two-dimensional fixed cylinder at Re=3900. y+ | St | $\bar{Nu}$ |  Cd | No. of cells
---|---|---|---|---
0.99 |  0.236 |  44.718 |  1.628 |  17814
0.69 |  0.233 |  43.918 |  1.640 |  25242
0.68 |  0.236 |  43.730 |  1.631 |  29516
0.67 |  0.237 |  43.395 |  1.628 |  39578
Table 2: The result of time-step size independence test for flow around two-dimensional fixed cylinder at Re=3900. St | $\bar{Nu}$ |  Cd | Time step
---|---|---|---
0.254 |  43.772 |  1.631 |  0.01
0.236 |  43.730 |  1.631 |  0.001
0.233 |  43.795 |  1.629 |  0.0001

###  4.2 Fixed cylinder

In this section, validation of the present obtained results for the fixed
cylinder are demonstrated. For this purpose, the results are validated
comparing the calculated pressure coefficient with the data reported in [31].
Also, the validation of the thermal part is performed via comparing the
obtained local Nusselt number distribution with the available data reported in
[32]. In [31], the flow around the cylinder in subcritical regime is simulated
utilizing the large eddy simulation method. The results obtained for the
pressure coefficient are, respectively, validated and verified with the
reported findings in [31] and [33].

Figure 3 displays this specific comparison. As it is clear in Fig. 3 a, the
difference between all three graphs is negligible at the separation point and
it gets larger going farther from this point. At the angles close to the
backward stagnation point, the calculated pressure coefficient in two-
dimensional cases deviates from the experimentally obtained pressure
coefficient presented in [33]. The numerical analysis predicts that the
pressure coefficient decreases when getting closer to the backward stagnation
point, while the experimental study presents a steady trend. This deviation
has been also reported and more, justified by Mittal [34]. In experiments and
the three-dimensional simulations, the cylinder side walls or say the end
plates, make the formed vortices obliqued or in other words cause the
formation of non-parallel vortices and also their exitance from the
longitudinal direction of the cylinder, the phenomenon which do not occur in
two-dimensional cases. Actually, in absence of curved vortices; the small-
scaled effects, which only exist in three-dimension, still reduce the minimum
values in the lift and drag oscillations. In three dimensional experiments and
simulations, the presence of oscillation along the centerline of the cylinder
and also the existence of vorticity component along the flow, results in
notable difference in the Reynolds stress and all components of the stress in
flow wake. This effect in turn, affects the pressure distribution, which
itself impresses the drag force.

Figure 3: Validation of (a) the pressure coefficient distribution, (b) the
local Nusselt number of the fixed cylinder in turbulent flow, when Re=8290.

Furthermore, to confirm the accurateness of the calculations related to the
heat transfer around the cylinder, the obtained local Nusselt number
distribution about the cylinder has been compared with that of calculated in
[32]. In this work, the heat transfer around the cylinder has been studied
numerically. The local Nusselt number for Re=8290 has been compared with the
experimental results of Schmidt and Wenner [36]. In Fig. 3 b, these two
previous findings as well as the present result for the local Nusselt number
are shown. As Fig. 3 b confirms, the numerical results of the present study
together with the numerical calculated Nusselt number of [32], slightly differ
from the experimental results reported in Wenner [36] just after the
separation point and in the positions near the rear side stagnation point of
the cylinder. This discrepancy is attributed to the presence of the three-
dimensional effects in experiments. This difference is much less recognized in
positions placed before the separation point. More particularly, near the
frontal stagnation point, the obtained results are close to the experimental
data.

###  4.3 Oscillating cylinder

In this section, the obtained results for the pressure coefficient and the
local Nusselt number are presented and compared with previous data for
oscillating cases. The pressure coefficient is validated using the available
data in [37], where the big vortices around an oscillating cylinder are
numerically investigated in two and three dimensions. Fig. 4 demonstrates the
available data for pressure coefficient distribution for two-dimensional cases
at Re=24000 [37] and the present acquired pressure coefficient. As it is
obvious, the error between two plots is about 0.2 before and after the
separation point.

Figure 4: Validation of the pressure coefficient distribution around
oscillating cylinder in turbulent flow at Re=24,000. Figure 5: Validation of
the local Nusselt number distribution around the oscillating cylinder in
Re=200 for (a) t=T/4, (b) t=3T/4.

In order to check the accuracy of the dynamic mesh part of the solver, the
obtained local Nusselt number distribution around an oscillating cylinder has
been compared with the results calculated in [8]. In this study, the flow
structure and the heat transfer about the oscillating cylinder is investigated
where the effects of the Reynolds number, and the amplitude have also been
inquired. The available reported data and also our obtained results at four
different times during one period of oscillation at Re=200, A/D=0.4,
$f^{*}$=0.2, are displayed in Fig. 5. At t=3T/4 and t=T/4, the cylinder is,
respectively, at its upper and lower positions during a cycle. Consequently,
the difference between the plots of the Nusselt number distribution is
increased up to 8$\%$, while this difference for t=T and t=T/2 when the
cylinder is placed in the origin of oscillating is less than 0.2$\%$.

##  5 Results and discussion

Table 3 presents list of details of different cases studied in this section.
Input parameters include position of slot, breadth of slots, number of slots,
amplitude of oscillation, height of fins, column of oscillatory excitation,
being slotted, inserting fins, single/bank of tubes.

Table 3: Different case studies and the value of related variables. Cases | Fin | Slot | Oscillation |  Slot position (r∗) |  Slot breadth (e∗) |  No. of slots (N) |  Amp. of Osc. (A∗) |  Height of Fin (H) |  Oscillating column
---|---|---|---|---|---|---|---|---|---
Fixed cylinder without fin |  |  |  | - | - | - | - | - | -
Fixed cylinder with fin without slot | ✓ |  |  | - | - | - | - | 0.75 | -
Fixed cylinder with slotted fin | ✓ | ✓ |  |  0.05-0.15 0.25 |  0.01-0.02-0.04 0.06 | 1-2-3 | - |  0.65-0.75 0.85 | -
Oscillating cylinder with slotted fin | ✓ | ✓ | ✓ |  0.05-0.15 0.25 |  0.02-.0.04 | 3 | 0.2-0.4-0.6 | 0.65 | -
Fixed bundle of tube without fin |  |  |  | - | - | - | - | - | -
Oscillating bundle of tube without fin |  |  | ✓ | - | - | - | 0.4 | - | 1-2-3-4
Oscillating bundle of tube with slotted fin | ✓ | ✓ | ✓ |  0.05-0.15 0.25 | 0.02-0.04 | 3 | 0.4 | 0.65 | 1-2-3-4

###  5.1 Fixed finned cylinder

Figs. 6 a and b demonstrate the pressure coefficient and the local Nusselt
number distributions over the finned cylinder without slot, for different ”s”
parameter, which is the coordinate attached to the surface of the cylinder.
The coordinate ”s” starts from s=0 located at the tip of the fin number 1, and
continues up to s=3.02 at the tip of the fin number 4. s$<$1.5 corresponds to
the low-pressure region, or say the region in the right-hand side of the fin
number 2.

In order to better discuss the pressure coefficient distribution and the
Nusselt number diagrams, points A to G are considered at the upper half of the
finned tube, and their corresponding points with the same names specified at
the ”s”-axis. In more details, point A is placed at the tip of the fin$\\#$1,
which is displayed at position s=0. B denotes fin$\\#$2 connection to the
surface of the cylinder, and corresponds to s=0.37. Also, points C and E are
the left and right roots of the fin number 2 respectively, and are shown at
s=1.13 and s=1.18. The D point is located at the tip of the fin$\\#$2 and
divides the plot into two parts. Points F and G also correspond to fin$\\#$4
and are placed, respectively, at the jointing points of the fin to the
cylinder and the tip.

As it is obvious in Fig. 6, the points being implemented subsequently at
s=2.65 and s=3.02, are presented in both the ”s”-axis and the schematic plot.
The time-averaged pressure for all the points with s¡1.5, except the tip of
fin$\\#$1 where the pressure is slightly higher, is approximately uniform. The
pressure coefficient in the left region of the fin$\\#$2, from point A to D is
positive, while it is uniformly negative in the right side of the point D. It
takes the lowest value in the point close to the tip of the fin$\\#$4, or say
G. The pressure coefficient suffers a sharp reduction from the positive value
to the negative one at the tip of the fin$\\#$2\. The pressure difference
between the two sides determines the value of the pressure (form) drag
coefficient. As much as the pressure difference grows, the pressure drag
coefficient gets larger. Moreover, the drag coefficient of the cylinder with
four fins, Cd=3.369, is found to be almost two times larger than that of the
cylinder without fin, Cd=1.631.

In Fig. 6 b, the Nusselt number distribution versus ”s” has been illustrate.
The maximum value of the Nusselt number occurs for the stagnation point A,
placed at the tip of the fin$\\#$1\. The value of Nu decreases going from
point A to the point B. In vicinity of the joint point of the fin to the
cylinder, around point B, a relative maximum of the Nusselt number is
conspicuous. The points denoted by RPi with i being 1 to 9 and marked in Fig.
6 b, are the points for which in closeness, the relative maximum of the Nu
occurs. Each of these points, is representative of the reattachment point that
is the point where the dividing streamline reattaches to the cylinder wall.
Also, Zdanski _et al._ [38] mentioned this Nu augmentation near the
reattachment points, studying the backward facing step case with considering a
turbulence promoter. This Nu increase is attributed to the heat transfer rate
augmentation caused by the increment of the turbulence diffusion mechanism.

Figure 6: (a) The pressure coefficient distribution and (b) the local Nusselt
number around the finned cylinder without slot when H=0.75.

The time-averaged vorticity field is presented in Fig. 7. The points denoted
as RP can be also seen in Fig. 6 b. As it is recognized, the flow detaches
from the surface at the sharp tip of the fin number one, reattaches to the
cylinder at point RP2. Fluid flows over the surface, and again separates from
the cylinder surface. The separated flow at point RP4, attaches to the
fin$\\#$2\. In the right half of the body, the separated flow forms a vortex
at the sharp tip of the fin$\\#$2\. The sharp tip of the fin$\\#$4 creates
another bubble. The separated flow at the tip of the fin reattaches to the
cylinder surface at point RP7. Similar phenomenon takes place in the
neighborhood of points E and F in smaller scale. In vicinity of the mentioned
points, a local maximum for the Nusselt number is detected.

The local Nusselt number distribution presented in Fig. 6 is similar to the
one reported in [13]. In the mentioned study, the fins are considered
thicknessless. The Nu distribution is presented versus the angle on the
surface of the cylinder. As results in [13, 20] suggest, the Nu on the
cylinder surface increases from the connection point of the fin to the
cylinder. Reaching the relative maximum, it obeys the reduction trend until it
catches the connection point of the next fin to the cylinder. Just before
point C, a relative maximum appears in Nu distribution owing to the existence
of the point RP3. The Nu on the surface of the Fin$\\#$2, increases from the
point C that takes its highest value in the tip of the Fin.

In vicinity of the fourth reattachment point, a secondary extremum is apparent
in the Nu distribution. In tip of the Fin$\\#$2, due to the compression of
streamlines, the fluid takes a high velocity and the Nu increases. In other
side of this fin, from the tip of the fin to its connection point to the
cylinder, the Nu follows a decreasing trend, and due to the RP5 existence, a
relative maximum of the Nu is conspicuous. On the surface of the cylinder,
between fins$\\#$2 and $\\#$4 or say from point E to the point F, there exist
three reattachment points in the vicinity of which the relative maximums are
recognized. Generally, between the points E to F, first the Nu increases and
then it decreases until the root of the Fin$\\#$4\. Also, the growing trend of
the Nu is evident over the Fin$\\#$4\. The convection heat transfer
coefficient augments with increasing the vorticity due to the higher mixing of
the swirling flow.

Figure 7: Distribution of reattachment points in time-averaged vorticity field
around the finned cylinder without slots.

The vorticity contour around the finned cylinder without slot during the half
period, has been presented in Fig. 8. In the second half period, a similar
happening occurs in the lower half of the body. As it is recognizable, the
flow separates at the tip of fin number one. At this point the value of the
vorticity is high. The vortex resulted from this separation, affects the
Nusselt number on the surface of the Fin$\\#$1\. Specially near the
reattachment point that is close to the point B, the Nu increase is evident.
After flowing the fluid on the surface, it separates from, and subsequently
attaches to the fin number two. This leads to the formation of a region with
low vorticity in vicinage of the point C and Nu decreases in this zone.

In the right side of the fin number two, the flow structure is more complex.
Several vortices are created in the region between the fins$\\#$2 and $\\#$4,
which affect the flow field and consequently the heat transfer. The flow
separates at the sharp tip of the fin$\\#$2 and a vortex is formed in the
downstream. As Fig. 8 b suggests, this separated vortical flow hits the fin
number 4, and owing to the existence of the sharp point on the tip of this
Fin, separates from it and another vortex is created. Indeed, Fig. 8 b
presents the creation of another vortex simultaneously with this vortex, on
the cylinder surface. The two vertices grow together and each one affects the
thermal condition on a part of the surface of the body. Figure 8 c presents
the growth of these two vertices. The formed vortex at the tip of the
fin$\\#$4 influences the flow pattern over the fin and the cylinder surfaces
in the vicinity of point F.

Figure 8: The vorticity contour around the finned cylinder without slot during
the half period when (a) t=0, (b) t=T/8, (c) t=T/4, (d) t=3T/8, and (e) t=T/2.

###  5.2 Fixed slotted finned cylinder

As mentioned in the previous section, adding fins to the cylinder makes the
drag coefficient to increase and the Nusselt number to decrease. In this
section, to equalize the pressure distribution around the fin, slots on the
vertical fins will be considered in which the flow can pass through. These
slots modify the drag coefficient by creating the connection between the high-
pressure and low-pressure sides. Also, the fluid flow passing through these
slots changes the flow structure in the area between the fins$\\#$2 and
$\\#$4\. Effects of the parameters such as, the position of slot placement or
its distance from the surface (r∗), the width of the slot (e∗), the number of
slots on the fin (N), the length of the slot (H), and the oscillation
amplitude of the cylinder (A∗) on the drag coefficient, the Nusselt number,
the fin effectiveness, and the Nu-to-drag coefficient ratio will be discussed.

####  5.2.1 Position of the slot

Three different dimensionless distances of r∗=0.05, r∗=0.15, and r∗=0.25 for
the slot on the fin are considered. Also, the slot with four different widths
of e∗=0.01, e∗=0.02, e∗=0.04, and e∗=0.06 is located individually in the
mentioned positions, r∗. Fig. The drag coefficient versus r∗, has been plotted
in Fig.9 a. As it is obvious, in all cases of the cylinder with the slotted
fin, the drag coefficient is less than that of the one with a fin without
slot. For e∗=0.01, the drag coefficient does not significantly differ relative
to its value for the finned cylinder without slot. In this case, the drag
coefficient remains almost constant by increasing the r∗ and it does not
matter, how far the slot is from the cylinder. Also, when e∗=0.02, a slight
reductive trend relative to the r∗ increment is seen. On the other hand, the
drag coefficient enhances when r∗ increases, for e∗=0.04 and e∗=0.06. The drag
coefficient presents maximum reduction of 33$\%$ relative to its value for the
finned cylinder without slot. This maximum reduction occurs, for a slot with
e∗=0.06, placed in the nearest distance from the cylinder, r∗=0.05. In
general, when the slot is thin, the drag coefficient variation is less than
1.5$\%$, while for wider slots, this reduction is more prominent. Getting
closer to the cylinder surface, the drag coefficient decreases more.

Fig. 9 b demonstrates the Nu variation versus r∗ for different values of e∗.
For all values of r∗, placing slot on the fin, results in augmentation of Nu
in comparison to the finned cylinder without slot. The behavior of the Nu
versus r∗ for the two slots with smaller width of e∗=0.01 and e∗=0.02, is the
same, such that firstly it increases and then decreases. For two other wider
slots, the Nu continuously increases moving away from the cylinder surface.
The maximum Nu, which is 30.7$\%$ larger, belongs to the slot with e∗=0.02
placed at r∗=0.15, or say in mid areas of the fin. The plot presenting the
Nu/$C_{d}$ ratio is shown in Fig. 9 c. For all cases, an increase in value of
this ratio for slotted finned cylinders is obvious. When e∗=0.01, due to the
smooth variation of the Cd and Nu versus r∗, Nu/Cd ratio increases slowly.
Owing to the slighter change of the drag coefficient, the ratio Nu/$C_{d}$ for
e∗=0.02 obeys the same trend of the Nu, that is, augmenting at first and
secondly reducing. For slots with e∗=0.04 and e∗=0.06, both Nu and Cd increase
while r∗ enlarges. As the drag coefficient change is much stronger than the
Nusselt number variation, the ratio of these two dimensionless numbers
decreases with r∗ increasing. The largest value of the Nu/Cd, being 85$\%$
higher than the value for finned cylinder without slot, corresponds to the
widest slot placed in the closest distance from the cylinder surface.

The fin effectiveness, which is defined as the ratio of heat transfer from the
finned cylinder to the heat transfer through the cylinder without fin, versus
r∗ is indicated in Fig. 9. The Fin effectiveness is a function of Nu and the
heat exchange surface area. By keeping e∗ constant, the heat exchange surface
does not change as the slot moves away from the cylinder surface.
Consequently, the fin effectiveness follows a trend similar to the Nu changes,
for each e∗. The maximum value of the fin effectiveness for r∗=0.05 and
r∗=0.15 belongs to the slot with e∗=0.02, and also, in the furthest distance
from the cylinder surface, the highest fin effectiveness happens for e∗=0.04.
In best situation, at r∗=0.15, a slot with the width of e∗= 0.02, whose
effectiveness equals 1.894, increases the effectiveness of the system compared
to the non-slot mode by 31.5$\%$.

Figure 9: (a) The drag coefficient, (b) the Nusselt number, (c) the Nusselt
Number to the drag coefficient ratio, (d) the fin effectiveness versus r∗ for
case with different slot widths, e∗.

The drag coefficient variations result from the change of pressure coefficient
on the surface of the body. In order to investigate origins of the behavior of
drag coefficient variation with r∗, the distribution of the pressure
coefficient on the finned cylinder with slots, for different values of e∗ is
presented in Fig. 10. Fig. 10(a) shows the schematic of the cylinder with
slotted fins and the letters A to G on it in the insert. Also, on the ”s”
axis, the letters from A to G are seen, which are the points corresponding to
the schematic shape on the surface of the object. The point D placed in the
tip of the fin$\\#$2, divides the plot into two halves. The position of the
slots can be seen in both sides of the point D on the fin number 2. The points
C to D present the left side of the fin$\\#$2, where the slot in this area,
indicates the notch of fin surface on its left side. On the other hand, points
D to E is related to the right side of the fin and the presented slot shows
the notch of the right-side of the fin surface caused by it.

Fig. 10 also demonstrates the effect of the distance of the slot from the
cylinder surface, for slots with different widths. As it is obvious, placing
the slot somewhere from point C to D on the fin, leads to the increasement of
pressure coefficient in the left side of the part of the fin above the slot.
This is expected since this part of the fin acts like a separating object in
the flow path and a high-pressure stagnation point is formed on its front. Due
to vortices formed behind this area, a low-pressure area is created in the
backside.

Figure 10: The pressure coefficient distribution of the cylinder with slotted
fin configuration for different values of the slot distances from the surface
of the cylinder with slot thickness of: (a) e∗=0.01, (b) e∗=0.02.

Figure. 11 demonstrates the created stagnation point in the mentioned zone. As
the slot moves away from the cylinder surface, the high-pressure area becomes
smaller and its pressure decreases. It is also seen in Fig. 10 that the
farther the slot is from the point D and the closer it is to the points C and
E, the high-pressure area in front with higher pressure is wider. Furthermore,
as the slot moves away from the cylinder surface, the size of the vortical
area in the wake becomes smaller and the low-pressure area becomes closer to
the body surface, and consequently reduces the pressure on the right-side of
fin surface above the slot. At the lower part of the slot, the trend is
inverse of that of the above one. In other words, the closer the slot is to
the cylinder surface, a smaller and closer vortex is formed that results in
reduction of the pressure near this part.

Comparison of Fig. 11 a and Fig. 11 c presents, as the slot gets closer to the
cylinder surface, a more distinct vortex eye in the backward of the lower part
of the fin. On the other hand, when the slot is away from the cylinder and
closer to the tip of the fin, an evident vortex eye is apperceived at the
behind of the top of the fin. Due to the low pressure of the vortex eye, as
the slot moves away from the cylinder, the low-pressure vortex eye behind the
lower part of the slot moves away from the surface, and the vortex center
approaches the surface at the top of the slot.

Shrinkage of the high-pressure region above the slot on its left side and
increasing the pressure of the lower right side of the fin reduces the drag
coefficient. Fig. 10 a is correspondent to the slot with e${}^{*}=0.01$, which
has no effective effect on the drag coefficient distribution at any location,
r∗, it has been placed. In this case, the crossing fluid flow through the slot
is weak, has a low velocity and penetration capability, and can not affect the
wake behind the fin. Hence, the drag coefficient changes are low and almost
constant for all r∗. The drag coefficient follows the pressure distribution
and although it behaves nearly unchanging, but still its small maximum occurs
in the middle areas of the fin arounf r∗=0.15.

Fig. 10 b presents the results with e∗=0.02. In this case, the drag
coefficient is also affected by the pressure distribution on the fin surface,
except that at r∗=0.05, the pressure below the slot at right side of the
fin$\\#$2, decreases more in comparison to the one with e∗=0.01. This results
in augmentation of the drag coefficient compared to the two other r∗. Here,
also, the percentage of drag coefficient changes with r∗ is not high and is
below 1.5$\%$. For two wider slots, the trend of drag coefficient changes with
r∗ increasing, is incremental. For e∗=0.04 and e∗=0.06, the passing flow
through the slot, changes the pressure distribution over the cylinder and the
fin surfaces, as the jet crossing the slot is stronger and has a higher
penetration capability than the narrower slots. In this matter, not only the
passing flow rate through the slot is increased, but also, the crossing flow
from the slot has higher momentum. Hence, it affects the wake of the body and
also the formed vortices on its behind.

Figure 11: Formation of the stagnation point and the coherent vortex structure
near the above zone of the slot with e∗=0.02 for distances from the surface
equal to (a) r∗=0.05, (b) r∗=0.15, (c) r$*$=0.25.

Fig. 12 acts to guide our discussion by plotting the time-averaged streamlines
around the cylinder with different positions of the slots. In this case, due
to the small flow rate of the fluid passing through the slot, the formed jet
has lower momentum, and therefore has low penetration capability. Figs. 12 a-c
illustrating the streamlines when e∗=0.06 confirm that the closer the slot is
to the cylinder surface, the larger the size of vortices becomes. As the
length of the vortex formation area increases, and the low-pressure zone moves
away, the surface pressure augments, which itself results in drag coefficient
reduction.

Figure 12: The time-averaged variation of the streamlines around the cylinder
with slotted fin for e∗=0.06 with different values of slot distance from the
fin root, (a) r∗=0.05, (b) r∗=0.15, (c) r∗=0.25.

The Jet velocity profile passing through the slot is presented in Fig. 13.
According to this figure, as the slot gets wider, the velocity profile tends
from a parabolic state to a uniform profile, which marks the advent of
turbulent flow. This slot broadening enlarges the passing jet area and the
flow rate increases, as well, the passing jet becomes more turbulent with
higher momentum to penetrate. Consequently, the passing jet affects the
pressure distribution over a broader part of the surface. The R-$\omega$ model
in contrast to the large eddy simulation models, which has high power in
recognizing the local laminarization of the flow, are not able to detect local
laminar flow. Therefore, it is better to review the value of the turbulence
viscosity in order to check the accuracy of the results obtained for the jet
velocity profile passing through the slot for e∗=0.01 and e∗=0.02, where the
parabolic velocity profile is obtained.

Figure 13: The velocity profile of the jet passing through the slot for
different values of r∗, for (a) e∗=0.01, (b) e∗=0.02, (c) e∗=0.04, (d)
e∗=0.06.

The turbulent viscosity of the passing jet for different values of e∗ placed
in various positions is presented in Fig. 14. As Figs. 14(a) and (b) being
respectively for e${}^{*}=0.01$ and e${}^{*}=0.02$ suggest, despite the
parabolic velocity profile, which is the indicator of laminar flow, the
turbulence viscosity is not exactly zero. However, the turbulence viscosity is
very low for these cases, namely, being in order of magnitudes of $10^{-6}$
and $10^{{}^{-}5}$, accordingly, for e${}^{*}=0.01$ and e${}^{*}=0.02$. Also,
the turbulence viscosity for e${}^{*}=0.04$ and e${}^{*}=0.06$, which is
respectively shown in Figs. 14(c) and (d), has higher value in comparison to
the ones with narrower slots. This increasement shows the level of turbulence
of the flow. Also, it is seen that going away from the cylinder surface, the
velocity profile is deviated upward. This occurs because of the angle of the
input flow into the slot which increases with augmentation of r∗. This can
also be observable in Fig. 17. For instance, when e∗=0.04 and r∗=0.05, the
angle of the streamlines passing through the slot, is equal to $3.5^{0}$. When
the slot gets away from the cylinder, this angle increases such that for
r∗=0.5 and 0.25 it is, respectively, equal to 33∘ and 54.2∘.

Figure 14: Distribution of the turbulent viscosity of the jet passing through
the slot with different r∗ and different thicknesses: (a) e∗=0.01, (b)
e∗=0.02, (c) e∗=0.04, and (d) e∗=0.06.

Figure 15 demonstrates the effect of the slot distance from the surface of
cylinder (r∗) on the Nusselt number distribution versus “s” for slots with
different values of e∗. As it is distinctive, with placing the slot on the
fin, no notable change of the Nu distribution from A to B corresponding to the
fin$\\#$1, is felt. For all e∗s, with increasing r∗, the Nu distribution over
the cylinder surface from B to C tends to that of the finned cylinder without
slot. The slot at r∗=0.05 results in Nu augmentation on the surface specially
in the vicinity of Point C. The pressure difference between two sides of the
slot makes suction of the low-velocity fluid from the neighborhood of the
point C and its pumping to the other side of the fin$\\#$2 at proximity of the
point E. Thus, the dead fluid is sucked from the body surface and the fresh
high-momentum flow replaces. This substitution causes the convective heat
transfer enhancement and the Nu increment over the cylinder surface, and also
the fin$\\#$2, in closeness of the point C. In Fig. 12, as well, approaching
of the streamline to the point C with insertion of the slot at r∗=0.15 is
obvious.

Figure 15: The local Nusselt number distribution of the cylinder with slotted
fin configuration for different value of the slot distance from the surface of
cylinder, r∗, and with the slot thickness of (a) e∗=0.01, (b) e∗=0.02, (c)
e∗=0.04, (d) e∗=0.06.

The Nu distribution in the left side of the fin$\\#$2, that is from point C to
D; predicts as closer as the slot is to cylinder, a higher Nu near the lower
region of the slot. Also, as r∗ increases, Nu enlarges further at the top edge
of the slot but, in regions below the slot, it behaves like that of the fin
without the slot. The maximum local Nu around the slot at the left side of the
fin$\\#$2 occurs for r∗=0.15. Fig. 13 also demonstrates that the value of
maximum velocity for the slot placed at r∗=0.15 is slightly higher than
velocities obtained for other two cases. The flow in the slot, is a
Poiseuille-type flow which is created due to a pressure difference. The reason
for this little variation of the maximum velocity is the slight discrepancy of
the pressure between the two sides of the slot at different r∗. Section 5.1
mentions that the Nu in the middle area of the fin$\\#$2 and fin$\\#$4 is
under the influence of the complex flow with different vortices and their
interaction with each other.

Fig. 16 manifests the vorticity contours at different r∗s for e${}^{*}=0.01$
over a half-period, compared to the case of finned cylinder without slot. As
it is seen, the output jet from the slot does not affect the flow front and
the size of formation of the vortices is the same for all r∗. Also, it is
equal to that of the finned case without slots. Fig. 16 a, which is related to
the finned cylinder without slot, confirms that at point G and also on the
cylinder surface, between points E and F, two vortices are simultaneously
formed and grow.

Figure 16: The vorticity contour during a half-period when e∗=0.01, for
different values of r∗: (a) fined without slot, (b) r∗=0.05, (c) r∗=0.15, (d)
r∗=0.25.

The formed vortical flow, affects the vorticity field in the right side of the
fin$\\#$2 over the cylinder surface between the F and E points, and also on
the fin$\\#$4\. The formed vortex at point G, grows and causes fluid mixing
and convection enhancement on fin number 4 and also on the cylinder surface in
the vicinity of point F. The generated vortex on the cylinder surface also
grows and enhances the convective heat transfer on the cylinder surface in the
neighborhood of point E and also on the fin$\\#$2 surface. In general, for
e∗=0.01, the output jet from the slot, due to its low momentum, has low
penetration depth and does not have much effect on the vorticity field between
the fins number 2 and 4. The output jet strikes the formed vortex and prevents
it from having influence on the fin$\\#$2\. As this jet has no high
penetration capability, does not have notable effect on increasement of the
vorticity in these points. Consequently, the Nu at the right side of the
fin$\\#$2 and hence on the cylinder surface, specifically when r∗=0.05,
decreases relative to the case with the non-slotted fin. Going away from the
cylinder surface, the passing jet moves away from the formed vortices, and the
Nusselt distribution approaches the value of Nu for a fined cylinder without
slot.

The vorticity contours over a half-period for e∗=0.02, at different r∗s, are
compared with the results for a case with a non-slotted fin, in Fig. 17. In
this case, the passing jet from the slot has higher penetration power relative
to the one with e∗=0.01. When r∗=0.05, it prevents the formation of vortices
near the surface of the cylinder. Also, the fin$\\#$2 surface and the cylinder
surface in vicinity of the junction point of the fin$\\#$2 to cylinder, are
affected by the passing flow through the slot, and the vorticity inside these
regions increases. The vorticity augmentation results in local Nu
increasement, which is observable in Fig. 15. After the passing jet flows
through the slot, it detaches from the surface and reattaches the cylinder
near fin$\\#$4\. This pattern causes a region with low Nusselt number on the
cylinder surface. Hence, when the slot is in nearby the cylinder, in regions
near the fin$\\#$2 and fin$\\#$4, respectively, Nu increase and decrease, is
seen. Figure 17 indicates that when the slot is located in farther distances
from the cylinder surface, the vortex formed by the flow jet, gets nearer to
the fin$\\#$2 surface and by increasing the vorticity, the Nu grows in the
region over the slot. Moreover, when the slot keeps out from the cylinder, the
effect of the passing flow through the slot, on the vortices decreases and the
Nu value tends to that of the finned cylinder without slot. Additionally, Nu
variations for e∗=0.04 and e∗=0.06 is also seen. As the slot widens, the
penetration distance of the jet passing through the gap increases.

Figure 17: The same as 16 but for e∗=0.002.

####  5.2.2 Width of the slot

In Sec. 5.2.1, the effect of the slot distance from the base of the fin has
been investigated. Specifically, for each width, influence of three different
distances have been studied. In this section, four values of e∗ has been taken
for each of the three contemplated r∗, to investigate the influence of slot
width. Particularly, its effect on the drag coefficient, the Nu number, the
fin effectiveness, and the Nu to drag coefficient ratio has been investigated.
The impact of the slot width on various parameters has been illustrated in
Fig. 18.

Fig. 18 a presents the Nu variations versus e∗. As it is obvious, the behavior
of the Nu number changes with increasing e∗, is approximately identical for
both r∗=0.05 and r∗=0.15. First, the Nu augments by widening the slot. Then,
increasing e∗ from 0.02 to 0.04, it decreases. For the case with the widest
slot, the number of Nu increases slightly. For r∗=0.25, the longest distance
between the slot and the cylinder surface, the Nu variations with increasing
e∗ is at first increasing and then decreasing.

Fig. 18 b illustrates the drag coefficient variation versus e∗ for different
values of r∗. According to this figure, for all r∗ values, widening the slot,
causes a reduction in drag coefficient. Increasing e∗, the body front area
reduces and also the flow rate of the high-pressure passing fluid, enlarges.
This trend results in increasing the length of the vortex formation area in
the downstream. With vortices getting away from the walls, the pressure on the
surface increases and the form drag decreases.

The fin effectiveness versus e∗ which is somehow identical to the Nu
variations’ dependency, is illustrated in Fig. 18 d. The fin effectiveness
depends on both the Nu and the heat exchange surface between the body and the
fluid; with increasing e∗ when t∗¡e∗, or say when the width of the slot is
greater than the fin thickness, the heat exchange surface reduces and with the
change of e∗ from 0.04 to 0.06, despite the increase of the Nusselt number,
the fin effectiveness has decreased.

Fig. 18 d establishes the ratio of the Nu number to the drag coefficient that
increases with e∗ increment for all r∗. As the percentage of changes of the Nu
number with e∗ in comparison to that of the drag coefficient, is lesser, the
ratio Nu/Cd is more under the influence of the drag coefficient. Due to the
reduction of Cd, the ratio increases with the enlargement of the slot’s width.
As a conclusion, Fig. 18 states that as close as the slot is to the cylinder
surface, the e∗ dependency of the drag coefficient is more pronounced, and Cd
decreases with a steeper slope. Similarly, Fig. 18 emphasizes, the lesser the
r∗ is, the ratio Nu/Cd increases more with e∗ increasement.

Figure 18: (a) The Nusselt number, (b) the drag coefficient, (c) the fin
effectiveness, (d) Nu to drag coefficient ratio, for different slot distances
from the surface, r∗, and also for slots with different thicknesses, e∗.

The pressure distribution determines the form drag coefficient and Figs. 19
a-f reveal the effect of the slot width on the Nusselt number distribution and
the drag coefficient for different values of r∗. It is seen in Figs. 19 a, c,
e that for every r∗, with increasing the slot width, the pressure difference
between the two sides of the body decreases. This prompts the drag coefficient
reduction. The closer the slot is to the cylinder, the greater the pressure
coefficient variation with e∗ turns into. Hence, as the slot approaches
surface of the cylinder, the drag coefficient dependency on e∗ increases. It
also appears evident in Figs. 19 b, d, f that the effect of e∗ on the Nu
distribution greatly differs for various parts of the body. Due to the reverse
variations in diverse segments, with increasing e∗, the Nu variations are not
monotonous and determined. From point A to B, which corresponds to fin$\\#$1,
the slot widening has little effect on the Nusselt number variation. Only when
r∗=0.05, with increasing e∗, the Nu number decreases marginally.

Figure 19: (a) The pressure coefficient, and (b) the local Nusselt number
distributions over the cylinder with the slotted fin for different values of
the slot thickness and the different slot distances from the root, (a)
r∗=0.05, (b) r∗=0.15, (c) e∗=0.25.

The time-averaged temperature field for cases with slotted fins are
corroborated in Fig. 20. When the slot is close to the cylinder surface for
case a in the figure, as the slot widens, the fluid is sucked more strongly
towards the slot. So, the hotter fluid has no opportunity to flow over the fin
number 2, and isothermal lines are bent toward the root. This fact causes the
Nusselt number reduction before point C, which is already seen in Fig. 19 b.
For r∗=0.15 and r∗=0.25 cases that are respectively manifested in Figs. 19 d
and f, this effect is less conspicuous and the value of Nu before point C is
close to that of the finned case without slot.

The heated flow near the surface is sucked out and a cool fluid takes its
place. The wider the slot, the suction power increases, and the Nusselt number
on the surface augments. The Nusselt increment over the fin when the slot is
closer to the cylinder surface is larger. As the slot gets away from the
surface, the value of the Nusselt number between fins$\\#$2 and 4, gets closer
to that of the finned cylinder without slot. Fig. 20 denotes that for all r∗,
the wider the slot, the better the fluid at the left side of the fin$\\#$2 is
sucked and the cool fluid approaches it.

Also, in the right side of the fin$\\#$2, it is seen that with enlarging the
slot width, the leakage flow rate from the slot increases and more volume of
cool high-velocity fluid is sucked into this area. Hence, the temperature of
the fluid adjacent to the slot, decreases and with increasing the temperature
gradient between the surface and the fluid, the Nusselt number increases. In
Fig. 19 b, it is illustrated that at r∗=0.05, the Nu distribution in the
region between fins$\\#$2 and 4 is highly under the effect of the slot; the
wider the slot, a greater Nusselt reduction occurs in vicinity of the point F.

e∗ increasement results in higher penetration capability. The fluid with
higher momentum flows over the cylinder surface in closeness of point E, and
accordingly the Nu number increments in this region. This flow detaches from
the body and reattaches to the surface of the fin$\\#$4\. Here, a low-
vorticity region appears that weakens the convective heat transfer. Thereby,
near point F, the Nu reduction is seen. The passing flow through the slot
prevents vortex formation near the cylinder surface and in the widest case,
affects the vortex near point G and prevents its impressment on the body. In
this case, as the fluid has high momentum and attaches to the fin$\\#$4 in
vicinity of its end point, it increases the Nu at the end point G. Figures 19
d and f confirm that the Nu distribution in the region between fins$\\#$2 and
4, when r∗=0.15 or r∗=0.25 is to some extent similar to that of the finned
case without slot. Just a slight reduction is seen.

Figure 20: The time-averaged temperature field for cases with different e∗,
and also different values of the slot distance from the surface, (a) r∗=0.05,
(b) r∗=0.15, (c) r∗=0.25.

####  5.2.3 Number of slots

In Secs. 5.2.1 and 5.2.2, respectively, the effects of r∗ and e∗ on the drag
coefficient, the Nusselt number, the Nu to Cd ratio and the fin effectiveness
have been investigated. In this section, in order to investigate the influence
of the number of slots on the mentioned parameters, at each distance from the
cylinder surface, r∗, the width of the slot leading to the highest fin
effectiveness, is taken, in pairs for the two-slot case (N=2) and in threes
for three-slot case (N=3) are placed on the fin. At r∗=0.05 and r∗=0.15, the
slot with the width of e∗=0.02, creates the maximum effectiveness.

In this section, the slots with e∗=0.02 at r∗=0.05, and e∗=0.02 at r∗=0.15,
are named, respectively, S1 and S2. At r∗=0.25, the effectiveness of the fin
with widths of e∗=0.02 and 0.04 are, respectively, 1.845 and 1.846, and both
values are almost equal to each other. So, the slot width which causes a
larger Nu/Cd will be chosen. This ratio for e∗=0.02 and 0.04 is, subsequently,
12.794 and 13.728. Thus, the slot with the width of e∗=0.04 is selected and is
assigned as S3. For N=2, three different two-slot arrangements of these three
slots are contemplated. The case a is the one with S1 and S2 being both
created simultaneously on the fin. It means, two slots with the width of
e∗=0.02, are placed on the fin, at r∗=0.05 and r∗=0.15. For the case b, S1 and
S3 slots are generated over the fin. The slots with e∗=0.02 and e∗=0.04 are
simultaneously constructed on the fin, respectively, at r∗=0.05 and r∗=0.25.
In the third two-slot fin case, called as case c, S2 and S3 are placed,
consecutively, at r∗=0.15 and 0.25. For the three-slot case which is appointed
as case d, all the three slots of S1, S2, and S3 are created concurrently on
the fin. All arrangements are schematically presented in Fig. 1 c(i-iv).

Table 4 demonstrates the drag coefficient, Nu, the ratio Nu/Cd and the fin
effectiveness for the mentioned four cases (see Figs. 1 c (i-iv)). Among the
two-slot cases, the smallest drag coefficient belongs to the case c that has
the value 16.6$\%$ less than that of the case with the non-slotted fin. Also,
the maximum Nusselt number pertains to case a, which shows 55.8$\%$
increasement in comparison to the Nu for the case with the non-slotted fin.
The fin with two slots in all three cases, has a higher effectiveness relative
to the cases with one slot. Besides, the fin with three slots presents the
maximum effectiveness. In this specimen, the heat transfer is 2.553 times the
heat transfer around the cylinder without fins; while the cylinder with the
non-slotted fin has increased the heat transfer in comparison to the cylinder
without fin for 1.440 times. For the case d, where the three slots are
simultaneously placed on the fins number 2 and 4, the drag coefficient and the
Nusselt number with respect to all two-slot cases, consecutively, decreases
and increase. In case d with three slots, the drag coefficient as compared to
the case with non-slotted fin has been decreased for 23$\%$, and also the
Nusselt number has increased for 76$\%$. The maximum ratio of the Nu/Cd
corresponds to the case d, with the ratio being 2.3 times larger than the
value obtained for the cylinder with the non-slotted fin.

Table 4: The different case studies and the variations of related variables. No. of Slots | Cases |  Cd | Nu |  Nu/Cd | $\xi$
---|---|---|---|---|---
0 |  Fin without slot | 3.369 | 32.222 | 9.564 | 1.440
1 | Fin with s1 | 3.248 | 41.367 | 12.736 | 1.861
1 | Fin with s2 | 3.212 | 42.102 | 13.107 | 1.894
1 | Fin with s3 | 3.028 | 41.569 | 13.728 | 1.846
2 | Case (a) | 3.046 | 50.209 | 16.483 | 2.273
2 | Case (b) | 2.841 | 49.913 | 17.568 | 2.231

The pressure coefficient distribution and the local Nu variation for different
two-slot cases and also the three-slotted type configuration, have been
indicated in Fig. 21. As it is conspicuous, in all cases, at the left side of
the body, the pressure is reduced relative to the cylinder with non-slotted
fins. On the other hand, at the right side, where the pressure has been lower,
the pressure increases for all cases. Within the two-slot modes, the pressure
difference diminution between the two left and right sides, for case a is
lower than the two other cases and the plots for cases b and c are similar to
each other. Thence, the drag coefficient of case a is larger than that of
cases b and c, which only differ each other by 2.1$\%$. Fig. 21, showing the
local Nusselt number distribution for cases a to d, and the cylinder with the
non-slotted fins, confirms that the main increment in Nu through the addition
of a slot to the fin, takes place on the fin$\\#$2 surface. It means, the Nu
distribution over the fin number 2 for different two-slotted cases is a
combination of two modes in which the slots are located separately on the fin.
For instance, for case a, Nu at the top of S2 is similar to that of the mode
in which S2 is placed solely on the fin. Also, for points located underneath
S1, it is almost akin the case where S1 has been lonely placed on the fin.
Between these two slots, the Nu number increases relative to the mentioned two
cases.

Figure 21: The local Nusselt number variation for the cylinder with slotted-
fin configuration for different values of the slot distance from the surface
of the cylinder, r∗, and with the slot thickness of: (a) e∗=0.01, (b) e∗=0.02

Also, in case d that all three slots are created simultaneously on the fin,
some kind of superposition occurs inside the area between the slots and, the
Nu augments there. Since, the slots are placed all together, the suction of
the heated fluid adjacent to the left side of the fin$\\#$2, is amplified, the
cool fluid is replaced, and the value of Nu rises. On the cylinder surface,
between points B and C, the value of Nu for distinct cases are very close to
each other. The Nu increment is not seen at vicinity of point C for the case
c. This is due to the fact that S1, the nearest slot to the cylinder surface,
has not been placed on the fin for the case c. On the other side of the
fin$\\#$2, at the lower points of S1 and the upper points of S3, if these two
slots are existent on the fin, the Nu plots of the different cases coincide.
The value of Nu for this case is larger than the case without the slots. For
the case d, the value of Nu in the midst of the slots S1 and S2 is more
dominant than that of the case b and c, although in this region, the case a
has the maximum Nu.

####  5.2.4 Height of the fin

In this section, effect of the fin height on the drag coefficient, the Nusselt
number, the ratio Nu/Cd, and the fin effectiveness will be discussed. In all
previous studies, the height of fins has been taken to be constant and equal
to H=0.75. As mentioned in Sec. 5.2.3, the maximum value of the fin
effectiveness and also the ratio Nu/Cd occurs for the case where three slots
S1, S2, S3 are simultaneously placed on the vertical fins. In this section for
investigation of the effect of the height of the slotted fin, with keeping
constant the value of the dimensionless height of the horizontal fins equal to
H=0.75, the non-dimensional height of the fins number 2 and 3 is considered to
take the values, 0.65, 0.75, 0.85. The results correspond to the cases with
these three values of heights are given in Tab. 5.

Table 5: The calculated drag coefficient, Nusselt number, Nusselt to drag ratio, and fin effectiveness for different values of fin height, H. H |  Cd | Nu |  Nu/Cd | $\xi$
---|---|---|---|---
0.65 | 2.421 | 56.727 | 23.431 | 2.469
0.75 | 2.591 | 56.738 | 21.898 | 2.553
0.85 | 2.851 | 58.743 | 20.600 | 2.656

Obviously, increasing the height of the vertical fins, the body front’s area
against the flow augments and causes a wider flow wake and increases the drag
coefficient. Also as the H enlarges, the length of the region above S3, which
has the highest Nusselt number increases, and consequently results in an
overall Nu increment. The change of Nu with fin height augmentation from
H=0.65 to H=0.75 is almost 0.02$\%$. In spite of the drag coefficient and the
Nu enlargement, as the Nu variation is low relative to the drag coefficient
changes, the Nu/Cd ratio decreases. It is distinctive that the highest Nu/Cd
ratio occurs for H=0.65. Moreover, the height increment makes the fin
effectiveness increase. As previously mentioned, the fin effectiveness that is
proportional to the ratio of heat transfer coefficient of the finned cylinder
to the non-finned cylinder, depends on both Nu and the heat exchange area. As
the heat exchange area and the Nu increase with height growth, the fin
effectiveness also augments. In utilization of the fin in the tube bank, the
priority is given to the Nu/Cd ratio and the fin with three slots and H=0.65
is chosen. Because in the tube bank, due to the closeness of tubes, when the
length of the vertical fins gets longer, the fluid flow path in the lanes
between the tubes becomes blocked. So, although the H increment results in fin
effectiveness increasement, the shortest fin that has the highest Nu/Cd ratio,
is selected to be used in the bundle of tube simulations.

###  5.3 Oscillating bundle of tubes with slotted fins

In this section, the flow and heat transfer over bundle of tubes will be
studied. Specifically, using extended surfaces as a passive method and
oscillation as an active method are used to investigate the heat transfer
increment in an staggered orientation of bank of tubes with four columns. To
fulfill this aim, first, the effect of oscillating different columns of the
tube bundle will be explored, and then the consequences of adding fins to the
tube bank will be discussed. According to the previous studies [3, 4, 5, 6, 7,
8], when the lock-in phenomenon occurs, the heat transfer notably enhances. As
aforementioned, the lock-in takes place when the excitation frequency
coincides with the natural frequency. It was found [10] that according to the
amplitude-frequency plot, the lock-in does not occur for a specific frequency
range, but there exists an interval in this plot, where the lock-in takes
place. In fact, this plot can be divided to three parts called as the lock-in
region, the transition, and the non-locked mode. As lower as the oscillation
amplitude is, the lock-in phenomenon happens in a smaller interval around
$f^{*}$=1. By augmentation of the oscillation amplitude, this interval gets
larger and the lock-in becomes observable even at frequencies larger than one.

As suggested by Kumar _et al._ [10], in order to determine in which region of
the amplitude-frequency plot, the flow is placed, two conditions should be
scrutinized. The first requirement is that in the power density spectrum, the
dominant frequency should be equal to the excitation frequency. In other
words, $f/f_{0}$ should be around 1 and the pick in other frequencies, should
no longer exist and must be disappeared. Another condition is that the other
picks in this plot, if there are any, should be an integer multiple of $f_{0}$
and also $f/f_{0}$ ratio should be an integer number. If both conditions are
met, the lock-in has taken placed. If only the first condition is fulfilled,
the flow is in transition or the quasi lock-in mode. If none of the conditions
is satisfied, no lock-in occurs and the flow is in the non-locked zone.

####  5.3.1 Bundle of fixed naked tubes

In order to find the suitable amplitude and the frequency for oscillation of
different columns of the bundle of tubes, firstly, the natural frequency of
the tubes in fixed state should be known. So, the flow around the fixed tube
bundle at Re=5000 have been investigated. The correspondent results are
presented in Tab. 6. As it is seen, in general, each column presents a lower
drag coefficient relative to its upstream column. This drag coefficient can be
attributed to the influence of the upstream tubes in the flow path, which
results in formation of a low-pressure region in front of others. Also, the Nu
is found to decrease for the downstream tubes owing to the fluid heating after
passing the upstream tubes. Before the first column, the cool fluid hits the
tubes and heats up. Figures 22 a and b present the time-averaged pressure
field and temperature field, respectively, in which the pressure reduction and
the temperature increment are evident. The pressure difference between two
sides of the tubes around the first column is more dominant than that of the
second column and so on.

Table 6: The calculated drag coefficient, Nusselt number, Nusselt to drag ratio and Strouhal number, for each of the tubes in the naked bundle of tubes. Column number |  tube number |  Cd | Nu |  Nu/Cd | St
---|---|---|---|---|---
1 |  C11 | 1.725 | 45.244 | 26.228 | 0.242
|  C12 | 1.664 | 44.984 | 27.033 | 0.212
|  C13 | 1.744 | 44.912 | 25.752 | 0.242
2 |  C21 | 1.057 | 40.892 | 38.687 | 0.212
|  C22 | 1.077 | 41.696 | 38.715 | 0.212
3 |  C31 | 0.928 | 39.762 | 42.847 | 0.242
|  C32 | 1.428 | 48.137 | 33.709 | 0.212
|  C33 | 0.989 | 39.962 | 40.406 | 0.242
4 |  C41 | 0.538 | 31.966 | 59.416 | 0.212
|  C42 | 0.587 | 34.135 | 58.151 | 0.212
Figure 22: The time-averaged (a) pressure field and (b) temperature field
around the fixed bundle of tubes without fins.

By moving toward the downstream columns, both the drag coefficient and the
Nusselt number decrease. As the percentage of the drag coefficient changes is
greater than that of the Nusselt number, the ratio of these two dimensionless
numbers have been more under the influence of the drag coefficient and
consequently it increases. According to the data listed in Tab. 6, the
Strouhal number that indicates the dimensionless frequency of the vortex
shedding, has two different values. In all existing tubes except the upper and
the lower pipes in the first and the third columns, i.e., C11, C13, C31, and
C33, St is equal to 0.212. In order for the excitation frequency of the all
tubes to be close to its natural frequency, a number close to the average of
these two numbers has been selected as the forced oscillation frequency. The
forced frequency of the different columns of the bundle of tubes has taken to
be $f_{0}$=0.230. In this situation, for the tubes with St=0.212, the
dimensionless frequency is $f^{*}$=1.085, while for other tubes we have
$f^{*}$=0.95. Both values are close to one. The ultimate goal is finding the
best case from heat transfer increase and the pressure reduction point of
view, by using two methods of oscillation as an active method and adding the
fin as a passive method.

####  5.3.2 A single oscillating slotted finned cylinder

In the interest of finding the suitable amplitude for oscillation, a tube with
the slotted fin, which in Sec. 5.2 have been chosen to be utilized in a tube
bank, in different amplitudes and at $f_{0}$=0.230 is subjected to the forced
transverse oscillations. The finned cylinder oscillation have been studied for
three disparate dimensionless amplitude of 0.2, 0.4, 0.6, and the results are
presented in Tab. 7.

Table 7: The results obtained for the drag coefficient, the Nusselt number, and the Nusselt to drag ratio according to the dimensionless amplitudes for an oscillating cylinder with slotted fins; N=3 and H=0.65. A∗ |  Cd |  Nu |  Nu/Cd
---|---|---|---
0.2 |  2.085 |  59.221 |  28.403
0.4 |  2.357 |  71.150 |  30.187
0.6 |  2.659 |  77.307 |  29.070

The time-averaged pressure along with the streamlines for the cases with
oscillation with three different amplitudes are demonstrated in Fig. 23. It is
seen that, as the amplitude of oscillation increases, the low-pressure zone
beyond the finned cylinder expands and the pressure decreases in this region.
A vortical flow appears near the tip of fin$\\#$1 owing to the vertical motion
of this fin inside the fluid, and a low-pressure zone is created there. The
greater the amplitude of the oscillation, the lower the pressure of these
vortices becomes. On the other hand, with the oscillation amplitude increment,
the pressure at the left side of the fins$\\#$2 and 3 increases and the
pressure difference between the two sides of the finned tube augments, which
causes the drag coefficient increasement. Additionally, the Nu enlarges as the
oscillation amplitude rises. As the excitation frequency is the same for all
three cases, the relative velocity between the fluid and the surface grows and
the heat transfer rate increases with amplitude augmentation. Consequently, as
the drag coefficient and the Nu, both increase with amplitude increment, and
on the other hand they have a same order of magnitude variation, and due to
their inverse effect on the Nu to drag ratio, the maximum of this
dimensionless parameter is found for A$*$=0.4. In this circumstance, the Nu to
drag coefficient ratio is increased by 28.8$\%$ relative to the fixed case.

Figure 23: The pressure field and streamlines around the cylinder with slotted
oscillating fin with the frequency, $f^{*}$=0.230 for different values of the
oscillation amplitude: (a) A∗=0.2, (b) A∗=0.4, (c) A∗=0.6.

Figure 24 denotes the power density spectrum for the three oscillating cases
of the tube with slotted fins. For the finned tube, the dominant frequency is
equal to the excitation frequency, and the natural frequency disappears.
Another peak is seen at $f/f_{0}$=3, which is an integer multiple of the
excitation frequency. It should be mentioned that for three considered
oscillating amplitude, despite the fact that the excitation frequency is
almost 1.8 times the natural frequency, the lock-on phenomenon takes place. As
previously mentioned, Gau _et al._ [6] has also obtained that the lock-in, in
flow around the cylinder occurs not only in frequencies equal to the
excitation frequencies, but also happens when the frequency equates the
integer multiple of the natural frequency of the cylinder. In the present
study, it is attained that the same occurs for a finned cylinder.

Figure 24: The power density spectrum for fixed and oscillating cylinders with
the slotted fin.

####  5.3.3 Oscillating slotted-finned bundle of tubes

In order to investigate the effect of oscillation, different columns in the
bundle of tubes with the dimensionless oscillation amplitude of A∗=0.4 and the
frequency of $f^{*}$=0.230 have been subjected to the forced oscillation. The
drag coefficients of the existent tubes in the tube bank versus the number of
tubes in various oscillating situations are shown in Fig. 25. Generally, for
all four oscillating cases, the drag coefficient of the oscillating column is
increased relative to that of the fixed case. When the first column
oscillates, the drag coefficient for the mid tubes of this column with
$f^{*}$=0.95 grows and it decreases for the mid tube with $f^{*}$=1.085. The
same happens with the third column is oscillating, except that the drag
coefficient increases for all three tubes. It is needless to say that the mid
tube experiences less increment. When the second or fourth columns are
oscillating, the drag coefficient of these columns enlarges in comparison to
the fixed cases. In the case that the second column oscillates, the drag
coefficient of the third column decreases. Oscillation of each column reduces
the pressure on the low-pressure side behind it.

When the second or fourth columns are oscillating, the drag coefficient for
these columns is increased compared to the stationary state. If the second
column is oscillating, the drag coefficient of the third column is reduced.
The pressure reduction of the behind region results in drag coefficient
increment in the oscillating column. Additionally, due to the expansion of the
low-pressure region behind the oscillating column, the pressure in front of
the next column, placed in the downstream of the oscillating column, reduces.
Accordingly, the drag coefficient of the column, next to the oscillating
column, decreases. The oscillation of a column, also affects the pressure
distribution of the previous column, by slightly increasing the pressure of
the front side as well as the pressure at the behind of the previous one. As
the changes are minor, they have no significant effect on the drag coefficient
of the upstream column.

Figure 25 b compares the Nusselt number variation for different columns of
four variant oscillating cases with that of the fixed cases. As it is viewed
for the first and the third columns which have three tubes, in similarity to
the drag coefficient, the Nusselt number for the lateral tubes with
$f^{*}=0.95$ increases due to the oscillation. For the mid tube where
$f^{*}=1.085$, the Nu is not augmented. In the first column, Nu of the mid
tube has decreased and in the third column it is almost equal to that of the
fixed one. In cases with oscillating second or forth columns, Nu augments due
to the oscillation. Also, the second column oscillation results in a Nu
reduction in the third column. In a general sense, the trend of Nusselt
changes is similar to the drag coefficient variations. It is obtained that for
the tubes with excitation frequencies less than the Strouhal frequency, the Nu
increases while for the other tubes with excitation frequency larger than the
Strouhal frequency, the Nusselt number either decreases or does not change
relative to that of the fixed state.

Figure 25: The effect of the position of tube in a fixed bundle of tubes with
different oscillating columns on (a) the drag coefficient, (b) the Nusselt
number, (c) the Nusselt number to drag coefficient ratio, (d) the fin
effectiveness.

Figure 25 c demonstrates the Nu to drag coefficient ratio versus the number of
tubes for different oscillating cases along with the fixed state. The drag
coefficient and the Nusselt number of each oscillating column augments, but,
as the drag coefficient variation percentage is higher than that of the
Nusselt number, the Nu/Cd ratio for the oscillating column decreases. The Nu
variations occurs with a smaller slope and consequently, the Nu/Cd ratio
follows the upward trend. Finally, what is more important is the effect of
oscillation on the flow and heat transfer of the whole bundle of tubes.
Thereby, parameters should be defined to present the overall performance of
the tube bank for fixed and oscillating situations. The overall Nusselt and
the overall drag coefficient of the bundle of tubes are defined, respectively,
as the average of Nu and drag coefficient of all tubes. The ratio of overall
heat transfer of the oscillating bundle of tubes to that of the fixed one, is
equal to the average Nusselt ratio of all the tubes in the oscillating
situation to that of the fixed state. Further, the ratio of the drag force
applied on the oscillating tube bank to the force employed to the fixed case
as well as the average Nu versus average drag coefficient ratio for both
oscillating and fixed cases are investigated in Tab. 8.

Table 8: The averaged values of parameters corresponding to the heat transfer and the resistive force for an oscillating bundle of tubes with three-slotted fins and H=0.65. Case studies |  Nuavg |  Cd,avg | $\frac{\textnormal{Nu}_{avg}}{\textnormal{C}_{d,avg}}$ | $\xi=\frac{\textnormal{Q}_{Finned}}{\textnormal{Q}_{No-Fin}}$ | $\frac{\textnormal{F}_{d,Finned}}{\textnormal{F}_{d,No-Fin}}$
---|---|---|---|---|---
Fixed bundle of tubes | 41.169 | 1.174 | 35.067 | - | -
The oscillating first column |  41.114 |  1.171 |  35.110 | 0.999 | 0.997
The oscillating second column |  41.846 |  1.276 |  32.792 | 1.016 | 1.087
The oscillating third column |  42.469 |  1.294 |  32.819 | 1.031 | 1.102
The oscillating fourth column |  41.701 |  1.261 |  33.069 | 1.013 | 1.074

All oscillating cases except the first column oscillation, increase the
average Nu of the tube bank in comparison to the Nu of the fixed bundle of
tubes. The maximum heat transfer belongs to the third column oscillation that
presents 3.1$\%$ augmentation relative to the fixed case. Also, oscillation of
all columns excluding the first one, causes the applied drag force on the tube
bank grow. Oscillation of the first column of the tube bank, reduces the
average drag coefficient. The highest average drag coefficient occurs for the
case where the third column oscillates. It shows 10.2$\%$ increment relative
to the fixed one. The only oscillating case where both the Nusselt number and
the drag coefficient are reduced, is the one with first column oscillating. In
this situation, the maximum Nu/Cd ratio is found. In other oscillating cases,
the ratio Nu/Cd has been decreased in comparison to that of the corresponding
fixed cases. If only the heat transfer increment is under consideration, the
greatest heat transfer rate takes place for the case with the third column
oscillating. Normally, in order to optimize the heat exchangers, the heat
transfer rate and the frictional losses are considered simultaneously. The
maximum value of the Nu/Cd ratio exists for the case with first column
oscillating.

In the previous cases, the effect of oscillation on Nu and drag coefficient of
the tube bank has been investigated. It was shown that the lock-on phenomenon
occurs, for the column which is oscillating. In this section, for the sake of
examining the effect of adding the chosen fin in Sec. 5.1, this fin is
appended to all the tubes of the tube bank. The four preceding oscillating
cases are also considered for the finned bundle of tubes. The results appeared
in Tab. 9 confirm that the maximum Nusselt of the finned oscillating bundle of
tubes emerges for the case with the first column oscillating. In this case,
adding fins and forcing oscillation simultaneously, increases the average
Nusselt number of the bundle of tubes up to 32.5$\%$ relative to its value for
the fixed tube bank.

Table 9: The average values of the parameters corresponding to heat transport and fluid flow for an oscillating bundle of tubes with three-slotted fins and H=0.65. Case studies |  Nuavg |  Cd,avg | $\frac{\textnormal{Nu}_{avg}}{\textnormal{C}_{d,avg}}$ | $\xi=\frac{\textnormal{Q}_{Finned}}{\textnormal{Q}_{No-Fin}}$ | $\frac{\textnormal{F}_{d,Finned}}{\textnormal{F}_{d,No-Fin}}$
---|---|---|---|---|---
Fixed bundle of tubes without fin | 41.169 | 1.174 | 35.067 | - | -
Bundle of pipes with fin and first oscillating column |  54.564 |  1.552 |  35.157 | 2.527 | 1.325
Bundle of tubes with fin and second oscillating column |  48.933 |  1.346 |  36.354 | 2.226 | 1.054
Bundle of tubes with fin and third oscillating column |  50.435 |  1.397 |  36.102 | 2.261 | 1.079
Bundle of tubes with fin and forth oscillating column |  50.673 |  1.096 |  46.234 | 2.313 | 0.869

For the finned oscillating tube bank in similarity to the un-finned cases, the
highest drag coefficient occurs when the first column is oscillating. Besides,
the minimum drag coefficient pertains to the case in which the fourth column
oscillates. In this situation, the drag coefficient not only is not increased
but also is decreased by 13$\%$ relative to the fixed case without the fins.
It means that when the fourth column oscillates, although the front area of
the tubes against the flow is enlarged, but the average drag coefficient for
the overall tube bank decreases. According to the results demonstrated in Tab.
9, the largest heat transfer rate for the finned oscillating tube bank belongs
to the case with the first column oscillating where adding the fins to the
oscillating bundle of tubes increases the heat transfer rate up to 2.5 times
that of the case without fins. According to the results demonstrated in Tab.
9, the largest heat transfer rate for the finned oscillating tube bank belongs
to the case with the first column oscillating in which by adding the fin to
the oscillating bundle of tubes, the heat transfer rate, compared to the case
without fins increases for 2.5 times.

##  6 Conclusions

Turbulent forced convective heat transfer around a cylinder using the
k-$\omega$ SST turbulent model is two-dimensionally simulated. The effect of
adding oscillation and fins on heat transfer as an active and passive method,
respectively has been investigated. Further, creating slots on the fin as a
method to reduce the drag coefficient has been studied. In order to optimize
the thermal systems, the heat transfer rate and the frictional losses should
be contemplated concurrently. In this study, four radial fins, in pairs
perpendicular to the flow and in flow direction as an extended surface are
used to reinforce the heat transfer. The vertical fins make the surface of
body front against the flow wider and consequently increase the low-pressure
region behind the body. This fact augments the drag coefficient in comparison
to that of the fixed cylinder.

In order to reduce the drag coefficient, slots are placed on the vertical
fins. The cylinder case studies with different slot distance from the cylinder
surface, the width of the slot, the number of slots placed on the fins, and
the fin height and oscillating frequencies are studied. The results present
the drag coefficient reduction for all case studies. While the slot is thin,
no notable change in pressure distribution on the surface is created and the
drag coefficient does not depend much on the fin distance from the cylinder
surface.

For wider slots, the closer the slot is to the surface of the cylinder, the
lower the drag coefficient becomes. The slot has a different effect on the
local Nusselt number in different parts of the surface, hence, variation of
the overall Nusselt number does not have a constant trend by increasing the
fin distance from the cylinder. The maximum Nusselt number for a one-slotted
case occurs for the one with e∗=0.02 and r∗=0.15, which shows 30.7$\%$
increase relative to that of the finned cylinder without slots.

To find the suitable number of slots on fins, in each three considered cases,
the width of slot which brings the highest fin effectiveness is chosen and in
pairs or all three slots are placed on the fins. It is obtained that among the
different two- and three-slotted cases, the largest Nu/Cd ratio belongs to the
case study with three slots. In this case, the drag coefficient and the
Nusselt number, respectively, has been decreased by 23$\%$ and increased for
76$\%$ relative to the sample including the non-slotted fin. Also, the fin
effectiveness has reduced from 2.553 to 1.440. The results confirm that the
longer the vertical fins are, due to the greater local Nusselt number in
vicinity of the end of the fin and enlargement of the heat transfer surface,
the larger the overall Nu becomes. On the other hand, length increment of the
fins leads to the drag coefficient growth. So, the maximum Nu to drag
coefficient ratio belongs to the shortest fin.

Next, to enhance the heat transfer rate in the staggered bundle of tube
configuration, first, different columns are forced to oscillate. It is
acquired that the maximum Nu and the heat transfer coefficient appears when
the third column is coerced to oscillate where the heat transfer increases
almost by 3$\%$ relative to that of the fixed bundle of tube. Also, in all
oscillating cases except the one with the first column oscillating, the drag
coefficient increases. Due to the drag coefficient increment, the Nu to drag
coefficient ratio decreases with oscillation. Exclusively, with the first
column oscillating, the Nu/Cd ratio augments for 12$\%$ in comparison to the
fixed tube bank.

Secondly, to ameliorate the heat transfer in the tube bank, the fin is added.
For all tube banks with fins, the overall Nusselt number increases. The
highest Nu appertains to the case with the first column oscillating. In this
situation, the overall Nusselt number augments for 32.5$\%$, relative to the
fixed bundle of tubes without the fin. For all the samples apart from the case
with the fourth column oscillating, the fin accruing causes the drag
coefficient increase. When the fourth column of the bundle of tubes
oscillates, the drag coefficient decreases in comparison to both cases of
oscillating and fixed bundle of tubes without fins. Consequently, the maximum
Nu/Cd ratio occurs for the case with oscillating fourth column, which is
31.8$\%$ higher than the one for the fixed tube bank without the fin.

In general, the oscillation alone results in the overall drag coefficient
increment of the bundle of tubes and 3$\%$ increasement of the heat transfer.
This is while when both oscillation and slotted fins as a combined method are
applied, in spite of the drag coefficient increment, both the Nu/Cd ratio and
the heat transfer rate increase. At last, to optimize the functionality of
oscillating heat exchangers by increasing the rate of heat transfer, and also
the Nusselt number to drag coefficient ratio, adding the slotted fin is
suggested.

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