Vortex-Induced Vibration for Heat Transfer Enhancement

VORTEX-INDUCED VIBRATION FOR HEAT TRANSFER ENHANCEMENT

Junxiang Shi, Steven R. Schafer, Chung-Lung (C.L.) Chen* Department of Mechanical & Aerospace Engineering, University of Missouri, Columbia, MO, United States

*[email protected]

ABSTRACT A passive, self-agitating method which takes advantage of

vortex-induced vibration (VIV) is presented to disrupt the

thermal boundary layer and thereby enhance the convective

heat transfer performance of a channel. A flexible cylinder is

placed at centerline of a channel. The vortex shedding due to

the presence of the cylinder generates a periodic lift force and

the consequent vibration of the cylinder. The fluid-structure-

interaction (FSI) due to the vibration strengthens the disruption

of the thermal boundary layer by reinforcing vortex interaction

with the walls, and improves the mixing process. This novel

concept is demonstrated by a three-dimensional modeling study

in different channels. The fluid dynamics and thermal

performance are discussed in terms of the vortex dynamics,

disruption of the thermal boundary layer, local and average

Nusselt numbers (Nu), and pressure loss. At different

conditions (Reynolds numbers, channel geometries, material

properties), the channel with the VIV is seen to significantly

increase the convective heat transfer coefficient. When the

Reynolds number is 168, the channel with the VIV improves

the average Nu by 234.8% and 51.4% in comparison with a

clean channel and a channel with a stationary cylinder,

respectively. The cylinder with the natural frequency close to

the vortex shedding frequency is proved to have the maximum

heat transfer enhancement. When the natural frequency is

different from the vortex shedding frequency, the lower natural

frequency shows a higher heat transfer rate and lower pressure

loss than the larger one.

NOMENCLATURE H (mm) Height of the channel

L (mm) Length of the channel

W (mm) Width of the channel

D (mm) Diameter of cylinder

d (mm) Distance from cylinder axis to inlet

𝐮𝐟 (m/s) Fluid flow velocity

𝐮𝐠 (m/s) Mesh velocity

𝑢𝑖𝑛 (K) Inlet velocity

ρ (kg/m3) Density

𝐶𝑝 (kJ/kg·K) Specific heat capacity

K (W/m·K) Thermal conductivity

𝑇𝑖𝑛 (K) Inlet temperature

𝑇𝑤 (K) Wall temperature

𝑓 Frequency (Hz)

St Strouhal number, = 𝑓𝐷/𝑢

𝑅𝑒𝐷 Reynolds number, = 𝑟𝑢𝑚𝐷/𝑚

INTRODUCTION Single-phase based convective heat transfer has been

widely used in various engineering applications. In recent

decades much effort has been devoted to enhancing the thermal

and hydraulic performance of single-phase channel flow [1] by

considering different mechanisms. It has been considered that

the cooling capacity of the laminar channel flow is hampered

by the dominant diffusive transport within the thermal

boundary layer. Hence, most work has been devoted to

enhancing the heat transfer performance by emphasizing

boundary layer disruption. Encouraging results have shown that

surface interruption by means such as vortex generators [2-5]

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-37594

1 Copyright © 2014 by ASME

mailto:[email protected]

can significantly enhance the heat transfer coefficients.

However, surface interruption also increases the pressure drop

and thus requires higher pumping power.

Due to the development of microelectro-mechanical

systems (MEMS), heat transfer enhancement with active flow

control technologies have been implemented in the last decade.

Synthetic jet actuators were introduced to induce momentum

flux and improve the heat transfer coefficient of the heat sink

[6-8]. Ma and Chen et al.[9] and Yu and Simon et al.[10] used

vibrating fins actuated with piezoelectric material to break the

thermal boundary layer, and thereby enhance the convective

heat transfer coefficient. Yu and Simon et al concluded that the

turbulence induced by the tip of the plate substantially

increased the heat transfer and that the translational motion of

the plate performed better than the flapping motion. Air-jet-

impingement cooling [11, 12], which uses a high-pressure

pump to generate a jet of compressed air directed at a heat sink

surface, is also an effective method for reducing the thermal

resistance of the thermal boundary layer.

The phenomenon of vortex induced vibration (VIV)

has been widely investigated [13-19] due to its importance in

engineering. The results have demonstrated that the vortex

shedding can resonate with one of the fundamental natural

frequencies or harmonic components of the flexible structure.

However, most of the previous studies focused on the structures

themselves, and the effects of this passive flow-structure

interaction (FSI) on the heat transfer in the channels have not

been studied in theory and experiments. Heat transfer

enhancement by intensifying vortex interaction with the walls

has been numerically demonstrated by Fu and Tong [20] and

Celik and Raisee et al. [21]. In those works, a cylinder in a two-

dimensional channel is actively and transversely vibrated at low

Reynolds numbers (ReD<=500). The encouraging results prove

that the generated vortex dynamics can effectively disturb the

thermal boundary layer and increase the heat transfer

coefficient of the channel wall. In our recent work [22], we

presented the heat transfer enhancement of a channel flow

which exploited VIV. In this design, a cylinder with a flexible

plate was placed in a clean channel; the vortex shedding due to

the cylinder gave rise to the oscillation of the plate downstream.

The two-dimensional numerical results showed that an

enhancement of the average Nusselt number in comparison

with the clean channel and the channel with a stationary

cylinder can reach increases of up to 91.95% and 22.10% when

ReD=204.8. In our other work[23], we studied a three-

dimensional channel with a flexible cylinder, and the vortex

induced vibration of the cylinder was demonstrated to distinctly

enhance the convective heat transfer rate at different Reynolds

number.

In this work, we extend our investigation of vortex

induced vibration for heat transfer enhancement. A three

dimensional flow-structure-thermal interaction enabled model

is employed to study the influences of Reynolds number,

structural properties, and channel geometries on the heat

transfer enhancement by VIV.

MODEL DESCRIPTION As shown in the schematic in Figure 1, an insulated

cylinder with a diameter of D is placed at the centerline of the

channel with a height of H, a width of W, and a length of L.

The distance between the cylinder’s axis and the entrance is d.

The four sidewalls of the channel are heated at a constant

temperature. Table 1 lists the values of the dimensional

variables. Water is selected as the working fluid in the channel.

The thermal properties of water are assumed to be constant at

25ºC. The flow in the channel is assumed to be laminar.

L

d

D H

uin, Tin

x

y

W

z

y

Constant temperature boundary Tw

Top wall

Bottom wall

y

xz Side wall

Side wall

Figure 1 Schematic of the channel with a cylinder

Table 1 Dimensions (mm) of the channel with a cylinder

Length of channel - L 50

Width of channel – W 30

Diameter of cylinder - D 1.5

Height of channel – H 6

Distance from cylinder axis to channel entrance – d 5

The inlet velocities, inu , in the two cases are given as

0.075m/s and 0.1m/s, corresponding to ReD inu D ~ 126

and 168, respectively. The inlet temperature profile (equation

(1)) is given to study the influence on the thermal boundary

layer. Here, the mean temperature, mT , is 313.15K; the constant

temperature of the channel walls, wT , is 343.15K.

2( ) 1 ( / 0.5 )in w w mT T y HT T (1)

When vortices shed from the cylinder an asymmetric

pressure distribution develops between the upper and lower

surfaces of the cylinder, and a periodic so-called lift on the

cylinder is generated. This force will induce the vibration of the

cylinder in a cross-flow direction. In the studies of VIV, the


Strouhal number ( St ) ( St fD u ) is a dimensionless

parameter describing the relationship among the shedding

frequency, characteristic length, and velocity at free stream

conditions. The Strouhal number depends on the Reynolds

number, and its function has been well summarized in [24].

Numerical and experimental studies [14, 25, 26] have indicated

that the walls will suppress the vortex shedding when the

cylinder is placed in a confined channel. The block ratio of the

cylinder diameter over the height of the channel has a

significant influence on the aforementioned function if the ratio

is larger than 0.35. In this work the block ratio is 0.25, hence

the actual frequency of the vortex shedding is determined in the

simulation.

NUMERICAL METHOD As the cylinder vibrates the flow field changes and

affects the deformation of the cylinder in return. Therefore, a

two-way FSI enabled solver is indispensable when studying the

heat transfer enhancement by the vibration of the cylinder. This

two-way FSI solution requires two solvers for the

computational fluid dynamics simulation (CFD) and

computational structural dynamics simulation (CSD) separately.

In the present work, two commercially available solvers,

ANSYS CFX 14.0 and ANSYS Transient Structural Module,

are employed. In view of the frequency, the time step is tuned

to 0.001(s) to capture the periodic vibration with sufficient

resolution. An unstructured quadrilateral grid is used for the

three dimensional computational domain. Turek and Hron [27]

provided numerous benchmarks for validating the FSI solver.

This solver has also been validated in our previous work [22].

In addition, three grids are employed to identify the grid

independence. The element number of the structural domain is

~20,000. The element numbers of the fluid domain are

~150,000, ~300,000 and ~600,000, respectively. The difference

in the average outlet temperature and pressure loss between the

~300,000 and ~600,000 node domains is less than 1%. Thus, in

the following discussion, the gird with ~300,000 elements is

used, considering the computational time and accuracy.

RESULTS AND DISCUSSION In this section we investigate the performance of a

clean channel (channel I), a channel with a stationary cylinder

(channel II), and a channel with a flexible cylinder (channel III).

Two Reynolds numbers, ReD (126 and 168), are investigated

with two inlet velocities. We will study the influence of the

material characteristics of the cylinder and the channel

geometries. The local and average Nusselt numbers are defined

in equation (2) to evaluate the heat transfer rate, where T is

the temperature gradient of the fluid at the channel wall

boundary. Since the vortex dynamics mainly interact with the

vertical walls (top and bottom shown in Figure 1), the average

Nu is obtained by integrating the local value over the channel

length, width, and vortex-shedding period .

0

( , )

1 1[( )

2

( ) ]

w m

W L topwall

W L bottomwall

HNu x t T

T T

Nu Nu dxdzWL

Nu dxdz

(2)

Performance of VIV at different Reynolds numbers

Figure 2 Lifts on the stationary cylinder when Re 168D

4

2 4

2

1 1(A πD I, πD A g)

4,

64

n

n

s

K EIgf

W

(3)

Figure 2 shows the lift coefficients on the stationary

cylinder at Re 168D . The lift coefficients, LC , are calculated

by 20.5 refF u D , where F is the force (Lift or Drag) and refu

is the reference velocity, which is taken as 0.1 m/s. The

frequency is estimated as 17.9 Hz, corresponding to a Strouhal

number of 0.2685. In channel III, the flexible cylinder is

composed of linear material and is fixed at both sides, hence

the Young’s modulus of the material can be used to estimate

the natural frequency by equation (3) in [28]. Here, A is the

cross-section area of the cylinder, I is the area moment of

inertia, is the load per unit length including weight, nK is

the constant of mode taken to be 22.4, and g is the

gravitational acceleration. Density s and Poisson’s Ratio are

taken as 2343.8 kg/m3 and 0.3, respectively. The Young’s

modulus of the cylinder at Re 168D is approximated as

4.1×105 Pa, corresponding to a natural frequency of 19.6 Hz.

The natural frequency of the cylinder is 109.5% of the vortex

shedding frequency. When the VIV occurs in channel III, the

amplitudes and frequency of the lift coefficients change

relevantly due to the fluid-structure-interaction. It is seen that


the lift coefficient upon the vibrating cylinder is much larger

than that upon the stationary one. As the cylinder vibrates the

flow velocity at the moving direction of the cylinder increases

due to the smaller cross section, as a result, the static pressure

and lift coefficient on the cylinder surface increase. The

frequency, calculated from Figure 3(a), is about 18.5 Hz, which

is 103.4% higher than the original one. This can be attributed to

the larger natural frequency of the cylinder. Figure 3(b) shows

the displacement of the center point of the cylinder (x=5mm,

y=0, z=0). It is worth mentioning here that the cylinder has not

only a periodic vibration along the y direction but also periodic

movement along the flow (x) direction.

Figure 3 Displacement of the center point of the cylinder when

Re 168D (a) Y-displacement with time (b) x and y

displacement in one period

Figure 4 Vorticity contours (Left) and Temperature fields at xy

plane (z=0) (a) channel I (b) channel II (c-d) channel III, when

Re 168D

Since the maximum deformation of the cylinder

occurs at its center, where z=0, the temperature field and vortex

dynamics at the xy cross plane (z=0) and xz cross plane (y=0)

are used to investigate the fluid dynamics and heat transfer. In

order to investigate the vortex dynamics and corresponding

influence on the heat transfer, Figure 4 illustrates z-directional

vorticity contours and temperature fields in three channels at

plane z=0. In channel II the vortices shed downstream of the

cylinder and arrange close to the centerline of the channel. The

vortex shedding is damped along the downstream, which is

consistent with [21]. Previous work has asserted that the

channel with a cylinder performs better in terms of heat transfer

than the clean channel, as the generated vortex shedding

disturbs the development of the thermal boundary layer

downstream. In channel III, the vibrating cylinder at a higher

frequency reinforces the interaction between the vortex

structures and wall shear layers. As shown in Figure 4c and 4d,

the vibration of the cylinder results in an up-and-down motion

of the vortex cores and moves these cores closer to the wall

shear layers. It is noticed that the number of the vortex cores in

channel III increases downstream with the increase in strength.

It is considered that the vibrating cylinder breaks the vortex

structures into multiple vortex structures, which is beneficial to

the hydrodynamic mixing process and heat transfer. In channels

II and III the vortex dynamics facilitate mixing the hot fluid

from the walls to the center of the channel and vice versa. After

studying the temperature fields seen in Figures 4 there is no

doubt that channel III performs better than channel II. In

channel II the cold fluid is broken into smaller segments by the

vortex/boundary-layer interaction. These segments arrange like

a train similar to how the vortex cores do. In channel III, the

stronger interaction assists the colder segments in penetrating

the thermal boundary layers and moving downstream.

Obviously, these motions increase the interfacial area between

the hot and cold fluid, and eventually enhances the heat transfer

rate. The results re-confirmed the conclusion drawn in our

previous two-dimensional investigations [22].

Figure 5 illustrates the enhancement of the local Nu on

the top walls of channel II and III when Re 168D . Due to the

periodic motion of the cylinder, the local Nu on the whole

bottom wall is close to that of the top wall. In channels II and

III the local Nu at three downstream positions is improved by

the vortex shedding; including the middle of the channel and

two portions near the horizontal sidewalls. The enhancement of

the local Nu at several regions is more significant in channel III.

To explain this phenomenon, vorticity at four cross-sections

along the flow direction (x=15mm, 25mm, 35mm and 45mm)

are investigated in channel III, as shown in Figure 6. We can

clearly observe the movement in vorticity at 15mm as the

vibration of the cylinder. This cross-flow movement increases

the mixing of the fluid within this region and thereby the heat

transfer rate. The vorticity downstream also has similar

dynamics.


Figure 5 Local Nusselt number on the top walls of (a) channel

II and (b) III when Re 168D

Figure 6 Vorticity contours at four sections (x=15mm, 25mm,

35mm, and 45mm) in channel III when Re 168D

Figure 7 compares the local Nusselt numbers at the

same time step; the walls are located at the xy plane z=0. For

channel I, Nu decreases monotonically due to the increase in

the thermal boundary layer thickness and approaches the fully

developed value. For channels II and III, the disruptions of the

thermal boundary layer lead to a non-monotonic variation of

Nu. The vibrating Nu along the downstream can be attributed

to the disturbed thermal boundary layer shown in Figure 4. Nu

increases near the 5mm distance from the entrance as the

cylinder located here decreases the thickness of the thermal

boundary layer. When the flow passes the cylinder the

thickness of the thermal boundary layer increases, and

consequently Nu decreases. Downstream of the cylinder the

interaction between the vortices and the thermal boundary layer

begins to enhance Nu. For channel II, this enhancement

increases gradually until reaching the outlet. Therefore, two

mechanisms contribute to the heat transfer enhancement; one is

squeezing the thermal boundary by the structure, and the other

one is disrupting the thermal boundary layer through vortex

interaction. Therefore, the local Nu of channel III is expected to

be higher than that of channel II. It is seen that Nu increases

sharply just behind the cylinder and then decreases

monotonically. Even so, the downstream Nu is always bigger

than that of channel II. The strengthened vortex dynamics and

the interaction with the wall shear layer can explain the

variation of Nu. It is apparent that the maximum enhancement

occurs close to the cylinder.

Figure 7 Local Nusselt number at xy plane (z=0) when

Re 168D

Table 2 lists the average Nusselt numbers of the three

channels. When Re 168D , channel II increases Nu from

1.52 to 3.355 (+120.7%), and channel III increases to 5.078

(+234.8%). For channel I, the increasing Reynolds number

(inlet velocity) cannot improve the average Nu distinctly

because the heat diffusion within the thermal boundary layer

plays a dominant role in heat transfer. It is noticed that Nu decreases slightly as ReD increases in channel I, which can be

attributed to the given inlet temperature profile. It is reasonable

that channel III performs better than channel II at the three

Reynolds numbers in view of the previous discussion.

Table 2 Average Nusselt number in three channels (Percentage

in the bracket is heat transfer enhancement compared to

channel I)

ReD 168

I/ Nu 1.52

II/ Nu 3.355 (+120.7%)

III/ Nu 5.078 (+234.8%)

It is known that the disruption of the velocity

boundary layer will lead to a greater pressure loss throughout


the channel. Moreover, the drag due to the structure also

contributes to the pressure loss of channels II and III. Figure 8

shows the drag coefficients upon the cylinder in channels II and

III. It can be seen that the VIV increases the drag coefficients in

a manner similar to that of the lift coefficients. Therefore, it is

not surprising that channel III has a larger pressure loss than

channel II. To evaluate the pressure loss coefficients of

different channels, the average value over a period is taken into

account and listed in Table 3. The pressure loss coefficients are

computed by 20.5 refP u . Comparing the thermal and

hydraulic performances between channels II and III, it is seen

that channel III increases the pressure loss by 16.79%, while

simultaneously improving the average Nu by 51.4%.

Table 3 Pressure drop coefficients in three channels

(Percentage in the bracket is pressure loss coefficient compared

to channel I)

ReD 168

I / cP 1.58

II / cP 2.74 (73.4%)

III / cP 3.2 (102.5%)

Figure 8 Drag coefficients in channel II and III when Re 168D

Effects of material properties

Celik and Raisee et al. [21] assert that the thermal

performance has the maximum improvement when the

vibrating frequency is 75% of the vortex shedding frequency.

Meanwhile, the amplitude of the vibration could have

significant effects. In this work, the cylinder is self-agitated due

to the FSI. Therefore, the vibrating frequency and amplitude

not only depend on the vortex shedding process but also the

properties of the cylinder, including Young’s Modulus and

density. To study the effects of the characteristics of the

cylinder, the Reynolds number is set as 126. When Re 126D ,

the original vortex shedding frequency is 13.4 Hz. If the density

and Young’s modulus of the cylinder is 2343.8 kg/m3 and

2.30625×105 Pa, respectively, the corresponding natural

frequency of the cylinder is 14.7 Hz, which is close to 13.4Hz.

Then, the density is tuned to be 1042 kg/m3 and 4166.8 kg/m3,

as a result, the natural frequency is 22 Hz and 11.02 Hz. The

vertical displacements of the cylinder are compared in Figure 9.

The cylinder with a density of 2343.8 kg/m3 has the largest

displacement because its natural frequency is closest to the

original vortex shedding frequency. Due to the difference

between the original vortex shedding and natural frequency

increases, the cylinder has a smaller deformation which can be

attributed to the weakened resonance. The frequency in Figure

9 is 14.3 Hz, 13.8 Hz, and 11.9 Hz for the cylinder, which is

88.8%, 103.0% and 106.7% of the original vortex shedding

frequency. Apparently, the cylinder with the smallest natural

frequency has a greater impact on the oscillating frequency of

the cylinder. Here, the cylinder with the greatest natural

frequency has the smallest deformation. As such, it is

reasonable to consider that this cylinder will have the lowest

heat transfer enhancement.

Figure 9 Y-directional displacements of the cylinder with

different properties at Re 126D

Table 4 Average Nu and Pressure loss coefficients of the

channels with three cylinders at Re 126D

Nu p

I 1.561 1.068

II 3.038

(+94.6%)

1.751

(+63.9%)

III 3( )s kg m

2343.8

4.5671

(+192.6%)

2.03

(+90.0%)

III 3( )s kg m

1042 3.358 (+115.1%) 1.87 (+75.1%)

III 3( )s kg m

4166.8 4.108 (+163.2%) 1.86 (+74.2%)

Table 4 lists the thermal and hydraulic performance by

the average Nu and pressure loss coefficient. The cylinder with

the greatest natural frequency loses more heat transfer


enhancement than the others, but the corresponding pressure

loss is not the smallest one. On the contrary, the cylinder with

the smallest natural frequency has the smallest pressure loss

and an intermediate heat transfer enhancement. Therefore, it is

concluded that for the same Young’s modulus, the natural

frequency of the cylinder should be close to the original vortex

shedding frequency to reach the maximum heat transfer

enhancement, otherwise, it is better for the frequency to be as

small as possible.

Effects of channel geometries

It is noticed that the Young’s modulus of the solid

cylinder is much smaller than various metals and alloys. To

overcome this shortcoming, the cylinder may be designed as a

hollow tube. Meanwhile, several sorts of polymers can satisfy

different properties, for instance, the Young’s modulus of

polybutadiene is about 1.6x106 Pa, and that of polyurethane is

25x106 Pa. When the width of the channel increases, the length

of the cylinder extends, therefore the Young’s modulus of the

cylinder should increase at the same flow conditions. In this

part, the width is taken as 63mm. The Young’s modulus of the

cylinder is chosen as 4x106 Pa, the density is considered as

1200 kg/m3 and 900 kg/m3, and the consequent natural

frequency is 19.4 Hz and 22.5 Hz. The performance of both

cylinders is listed in Table 5. It can be found that the cylinder

with larger density and corresponding lower natural frequency

has higher thermal performance enhancement and lower

pressure loss than the other one.

Table 5 Average Nu and Pressure loss coefficients of the

channels with two cylinders at Re 168D

Channel I II III3( )s kg m

=1200

III 3( )s kg m

=900

Nu 1.54 3.31

(+114.9%)

4.56

(+196.1%)

4.44

(+188.3%)

p 4.23 7.58

(+79.1%)

8.76

(+107.1%)

8.89

(+110.2%)

CONCLUSION A cylinder which is self-agitated due to vortex induced

vibration is introduced to enhance the convective heat transfer

coefficient of the single-phase channel flow without any

additional power. A three-dimensional model with an FSI

enabled solver is used to demonstrate the performance of this

approach at different Reynolds numbers and in different

channels. Three channel configurations are studied: a clean

channel, a channel with a stationary cylinder, and a channel

with a flexible cylinder. The results demonstrate that the

channels with VIV perform the best at all of the flow

conditions as the periodic vibration of the cylinder strengthens

the vortex interaction with the walls, and thereby enhances the

mixing process. The properties of the cylinder, and different

Reynolds numbers and channel geometries are investigated. If

the Young’s modulus is the same, the VIV reaches the

maximum performance once the natural frequency of the

cylinder is close to the original vortex shedding frequency.

When the natural frequency is different from the vortex

shedding frequency, the lower natural frequency is better than

the larger one when considering hydraulic and thermal

performance. Therefore, the channel with the VIV is considered

to be a promising way to further enhance the heat transfer rate

of the single-phase channel flow with acceptable pressure

penalty.

ACKNOWLEDGEMENTS This research is sponsored by startup funding from University

of Missouri and Dr. Myers, Joseph at ARO under contract No.

W911NF-12-1-0147.

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Vortex-Induced Vibration for Heat Transfer Enhancement