effects of axis ratio on the vortex-induced vibration and

1 Copyright © 2014 by ASME

EFFECTS OF AXIS RATIO ON THE VORTEX-INDUCED VIBRATION AND ENERGY HARVESTING OF RHOMBUS CYLINDER

Li Zhang [email protected]

Key Laboratory of Low-grade Energy Utilization Technologies

and Systems of Ministry of Education of China(Chongqing

University) Chongqing , China

Heng Li [email protected]




Lin Ding [email protected]




ABSTRACT The vortex-induced vibrations of a rhombus cylinder are

investigated using two-dimensional unsteady Reynolds-

Averaged Navier-Stokes simulations at high Reynolds numbers

ranging from 10,000 to 120,000. The rhombus cylinder is

constrained to oscillate in the transverse direction, which is

perpendicular to the flow velocity direction. Three rhombus

cylinders with different axis ratio (AR=0.5, 1.0, 1.5) are

considered for comparison. The simulation results indicate that

the vibration response and the wake modes are dependent on

the axis ratio of the rhombus cylinder. The amplitude ratios are

functions of the Reynolds numbers. And as the AR increases,

higher peak amplitudes can be made over a significant wide

band of Re. On the other hand, a narrow lock-in area is

observed for AR=0.5 and AR=1.5 when 30,000<Re<50,000, but

the frequency ratio of AR=1.0 monotonically increases at a

nearly constant slope in the whole Re range. The vortex

shedding mode is always 2S mode in the whole Re range for

AR=0.5. However, the wake patterns become diverse with the

increasing of Re for AR=1.0 and 1.5. In addition, the

mechanical power output of each oscillating rhombus cylinder

is calculated to evaluate the efficiency of energy transfer in this

paper. The theoretical mechanical power P between water

and a transversely oscillating cylinder is achieved. On the base

of analysis and comparison, the rhombus cylinder with AR=1.0

is more suitable for harvesting energy from fluid.

INTRODUCTION Flow-induced motion (FIM) is a canonical problem of

fluid-structure interaction. Elastically mounted, rigid, bluff

bodies that are long in one direction perpendicular to a free

stream are susceptible to this phenomenon. The classic research

on FIM is that of a circular cylinder, elastically mounted,

immersed in a free stream. Much research has been made in this

field for the past decades. Generally speaking, vortex-induced

vibration is affected by a large number of system parameters,

which include the mass ratio, structure stiffness, system

damping, surface roughness of the cylinder, etc (1-5).

The oscillating amplitude is an extremely important

measurement, which can quantify the displacement response of

vortex-induced vibration (VIV). Moreover, the frequency is

another crucial measurement to describe the dynamic

characteristic of the oscillation system. As for a circular

cylinder subjected to flow-induced motion, several different

frequencies are defined to fully describe the phenomenon, such

as the natural frequency of the cylinder, the oscillating

frequency and the vortex shedding frequency. With the

increasing of flow velocity, the oscillating frequency will

remain around the natural frequency over a range of flow

velocity, this is called a lock-in area and the cylinder will

oscillate at a relatively high amplitude. An experimental study

of a circular cylinder with different mass ratios were conducted

by Govardhan and Williamson (7), the results show that the

oscillation frequency is a function that depends on the reduced

velocity.

On the other hand, in order to gain a better understanding

of VIV, Williamson and Roshko (9) studied the relationship

between the vibration amplitude and the vortex shedding mode.

Digital Particle Image Velocimetry (DPIV) was adopted in the

experiments to identify the vortex shedding patterns behind the

oscillating cylinder. Based on a large number of tests results, the

Williamson-Roshko wake mode maps (6, 9) were established.

Proceedings of the ASME 2014 Power Conference POWER2014

July 28-31, 2014, Baltimore, Maryland, USA

POWER2014-32156


Their findings suggest that the vortex shedding mode is closely

related to the vibration of the cylinder. Different wake modes

can be observed for a rigid cylinder with low mass and low

damping undergoing vortex-induced vibration, including the

“2S”, “2P”, “P+S” and “2P+2S”, etc(7-11). Here, the “S”

represents a single vortex and the “P” represents a pair of

vortices. Need to add that, more complicated wake modes could

be created in some circumstances, but these complex modes are

also made up of some basic modes.

Flow-induced motion is the product of the interaction

between fluid and the cylinder. This kind of interaction depends

upon two factors: the density and velocity of the fluid, the

stiffness and mass distribution of the structure. It’s significant to

note that the influence of the geometry of the oscillating

cylinder is one that has received little attention with regard to

VIV (12). Actually, geometrical features can have a significant

impact on the exciting force of flow-induced motion and the

study on this effect can offer some insight into the excitation

mechanism. After studying the flow-induced motion of two-

dimensional bluff body, Pakinson and Simith (13) pointed out

that the shape and size of a bluff body are the most important

parameters affecting the exciting force. Deniz and Staubli (14,

15) performed controlled oscillation experiments of

rectangular- and octagonal- section cylinders. The visualization

analysis of the flow field was adopted to show the process of

vortex shedding and collision. The research highlighted that the

length of the after-body (the body downstream of the separation

points of the shear layers) can have a substantial impact on the

oscillation response of the cylinder.

It can be inferred from the existing literature that data on

the flow over rhombus cylinders is limited, especially at high

Reynolds numbers. Meanwhile, research on the energy transfer

of rhombus cylinders has not been reported yet. Hence, the

focus of this numerical study is to investigate the influence of

geometry on vortex-induced vibration of rhombus cylinder and

to identify the passive geometric features that can be used to

generate large amplitude oscillations suitable to drive a

generator. In the present work, two-dimensional unsteady

turbulent flow over a rhombus cylinder with different axis ratio

(AR) was investigated numerically. The rhombus cylinder is

constrained to oscillate in the transverse direction. Numerical

simulations were performed for high Reynolds numbers ranging

from 10,000 to 120,000 in three different axis ratios, AR=0.5,

1.0 and 1.5. The influences of the AR on the amplitude and

frequency ratio, wake structure, and theoretical mechanical

power are obtained.

PHYSICAL SYSTEM The physical model considered in this paper consists of an

oscillatory system as depicted in Fig.1. The elements of the

oscillatory system are a rigid rhombus cylinder, K is the

stiffness of two supporting linear springs, and the system

damping C due to friction. The rhombus cylinders are

constrained to oscillate in the y-direction, which is

perpendicular to the flow direction (x). Three rhombus cylinders

with different axis ratio (AR=0.5, 1.0, 1.5) are considered for

comparison, where AR=L/D and L is the length of diagonal of

the rhombus cylinder parallel to the flow direction, D is the

length of diagonal vertical to the flow direction. The Reynolds

number is defined as Re =UD/ν.

Fig. 1 Schematic of the physical model

In this paper, the system parameters of the models in the 2-

D URANS simulation are listed in Table 1 and Table 2.

Table 1 Nomenclature

peakA Peak amplitude

rmsA Root-mean-square amplitude

D Length of diagonal vertical to the flow

direction

L Length of diagonal parallel to the flow

direction

Re Reynolds number

K Spring constant

,1/n n waterT f Natural period in water

U Flow velocity

systemC Structural damping

, / ( ) / 2n water osc af K m m System natural frequency in water

oscf Oscillating frequency of cylinder

dm Displaced fluid mass

a a dm C m Added mass

oscm Oscillating system mass

* /osc dm m m Mass ratio

Kinematic molecular viscosity

Density of the fluid

aC Added mass coef.

app:ds:parallel

app:ds:parallel


Table 2 Physical model parameters

Item AR=0.5 AR=1.0 AR=1.5

Diameter D [m] 0.0889 0.0889 0.0889

Length l [m] 0.9144 0.9144 0.9144

Oscillating system mass oscm [kg] 10.94 10.94 10.94

Spring const. K [N/m] 400 400 400

Damping systemC 14.98 14.98 14.98

Natural freq. in water ,n waterf 0.89 0.83 0.78

GOVERNING EQUATION In this section, the mathematical modeling for the fluid

dynamics is provided.

In this study, the flow is assumed to be two-dimensional

and unsteady, and the fluid is incompressible. The flow is

modeled using the Unsteady Reynolds-Averaged Navier-Stokes

(URANS) equations together with the one-equation Spalart–

Allmaras (S–A) turbulence model. The basic URANS equations

are:

0i

i

U

x

(1)

1( ) (2 )i

i j ij i j

j i j

U pU U S u u

t x x x

(2)

where, is the molecular kinematic viscosity and ijS is the

mean strain-rate tensor.

1( )

2

jiij

j i

UUS

x x

(3)

iU is the mean flow velocity vector. ij i ju u is known as

the Reynolds-stress tensor. The Boussinesq eddy-viscosity

approximation is employed to solve the URANS equations for

the mean-flow properties of the turbulence flow.

The dynamics of the rhombus cylinder is modeled by the

classical linear oscillator model.

( )oscm y cy Ky f t (4)

Here, oscm is the total oscillating mass of cylinder, c is

the damping of the system, and K is the linear spring

constant.

COMPUTATIONAL DOMAIN AND GRID GENERATION The computational domain is 50D×50D for the single

rhombus cylinder. As shown in Fig.2 (a), the entire domain

includes five boundaries: inflow, outflow, top, bottom, and the

cylinder. The cylinder is located in the center of the

computational domain. The inflow velocity is considered as

uniform and constant velocity. At the out flow boundary, a zero

gradient condition is specified for velocity. The bottom and top

conditions are defined the same as the inflow boundary. A

moving wall boundary condition is applied for the cylinder

when the cylinder is in FIM. In order to see the effect of the

axis ratio (AR) on the vortex-induced vibration of rhombus

cylinder, numerical simulations were done for high Reynolds

numbers ranging from 10,000 to 120,000 in three different axis

ratios, AR=0.5, 1.0 and 1.5.

(a)

(b)

Fig. 2 (a) Computational domain. (b) Sample of the

grid points for AR=1.5.

A grid sensitivity study was conducted on three different

grid levels for the rhombus cylinder with different AR. The

vibration displacement and lift coefficient were calculated for

comparison. For the present study, all three grids obtain similar

results. In view of the computer resources and computing time,

grid number 240×200 was chosen for all simulations. Fig.2 (b)

displays a sample of the grid points for AR=1.5.

RESULTS AND DISCUSSION Simulations were conducted to study the effect of axis ratio

(AR=0.5, 1.0, 1.5) on the fluid flow around a rhombus cylinder.

In particular, the correlations between the geometry and the

vibration response, the energy output have been analyzed.

The traditional measure of flow-induced motion response

has been the peak amplitude of oscillation. This measure admits

only harmonic body oscillations, meaning that the peak


amplitude and frequency of the oscillating cylinder are required

to fully describe the motion. It’s not a problem during fully

synchronized VIV. However, the vibration properties are not

clear when the flow is not completely synchronized to the body

motion. In order to get a better description of the amplitude

response of rhombus cylinder, the peak amplitude (peakA ) and

the root-mean-square (RMS) amplitude (rmsA ) are presented in

this paper.

The peak amplitude ratio (peakA /D) for the numerical

study are plotted in Fig.3a. Fig.3b presents a measure of the

RMS amplitude ratio (rmsA /D), and provides a more

meaningful measure of magnitude for asymmetric oscillations.

(a)

(b)

Fig. 3 (a) Peak amplitude ratio ( peakA /D). (b) RMS

amplitude ratio (rmsA /D).

Fig.3 shows the peak amplitude and the root-mean-square

amplitude of oscillation for all three geometries as functions of

Reynolds number:

a) AR=0.5: It should be noted that the peak amplitude ratio

and the RMS amplitude ratio follow the same trends as the

Re increases. A number of regions have been identified for

this cylinder in two different amplitude ratio curves. The

first region is the “initial” branch in VIV for Re<30,000.

The peak amplitude is below 0.1D in this region. Further

increasing Re sees the onset of the “upper branch” for

40,000<Re<60,000. The peak amplitude risen steeply from

0.08D to 0.68D when the Reynolds number reaches

40,000, then stays around 0.7D. With the increasing of

Reynolds number, the following region is identified as the

“lower branch” in which the peak amplitude gradually

dropped to about 0.4D and stays stable when Re increases

from 70,000 to 120,000.

b) AR=1.0: No obvious branches are observed in the

amplitude response of this rhombus cylinder. When Re

increases from 10,000 to 70,000, the peak amplitude rises

sharply from 0.05D to 0.97D. Then it stays around 1D and

the maximum amplitude 1.07D is obtained when

Re=120,000.

c) AR=1.5: No obvious branches are observed in the

amplitude response of this rhombus cylinder. As can be

seen from Fig. 3a, the peak amplitude increases

continuously with Re in the whole Re range. The

maximum amplitude 1.52D is achieved when Re=120,000.

However, the RMS amplitude has a different changing

trend, this difference indicates that the oscillating loses its

periodicity. Fig. 3b shows that the RMS amplitude stays

around 0.5D when Re>60,000.

Fast Fourier Transform (FFT) is used to process the

simulation records for each run and for each cylinder. The

major harmonic frequency of oscillation for each cylinder is

non-dimensionalized by the corresponding system natural

frequency in water,,n waterf . The frequency ratio is plotted versus

Reynolds number in Fig.4.

Fig. 4 Frequency ratio ( ,/osc n waterf f )

a) AR=0.5: As shown in Fig.4, The frequency ratio quickly

rises to 1.1 when Re reaches 30,000. With the increases of

Re, a narrow lock-in area is observed for

30,000<Re<50,000, which corresponding to the “upper

branch”. Following the lock-in area, the curve shows a


nearly constant slope when Re>50,000. Eventually, the

maximum frequency ratio 2.84 is achieved at Re=120,000.

b) AR=1.0: The frequency ratio of this cylinder

monotonically increases at a nearly constant slope in the

whole Re range. It's important to point out that the

frequency ratio curve of AR=1.0 keeps above the other

two curves when Re>50,000. The maximum value reaches

up to 4.1 at Re=120,000.

c) AR=1.5: As the Re increases to 30,000, the frequency ratio

reaches 0.98. After Re=30,000, frequency ratio stabilizes

around 1.1 for a narrow Re range, with the oscillation

frequency of the cylinder being very close to the system

natural frequency. When Re>60,000, the frequency ratio

increases sharply to relatively high value.

In fact, amplitude and frequency are integral properties of

the fluid-structure dynamics. The differences in vortex shedding

patterns are caused by the interaction of vortices being shed

from the sharp corners (including the leading edge and the

trailing edge) of the cylinder. To get a better understanding of

the relationship between the oscillation response and the vortex

shedding process, the vortex structures around the cylinder are

presented.

The simulation results of three typical Reynolds numbers

are presented. The vortex structure for each rhombus cylinder at

Re=30,000, Re=60,000, Re=90,000, and Re=120,000 are

presented in Figs.5-8, respectively.

The most striking feature of rhombus cylinders is the sharp

trailing edge can have a significant impact on the wake response,

and the vortex pattern becomes more complex when the AR

increases. It's important to note that the vortex shedding mode is

always 2S mode in the whole Re range for AR=0.5. However,

the wake patterns become diverse with the increasing of Re for

AR=1.0 and 1.5.

As for AR=0.5, the size of the trailing edge is relatively

small and the trailing edge has slight effect on the interaction

between the two shear layers. The two separating shear layers

could interact with each other freely, the vortex formed on one

side of the cylinder will be chopped by the shear layer from

another side of the cylinder. In this process, “2S” mode is

formed. However, the impact of the sharp trailing edge can no

longer be ignored for AR=1.0 and AR=1.5. The vortex

formation is influenced by the interaction between the shear

layer and the sharp trailing edge. As can be observed in the

images of vorticity, the sharp trailing edge delays the interaction

of the separation shear layers and snips the vortices formed

from the shear layers. For the “2P” mode, both forming vortices

are snipped by the sharp trailing edge. During the development

of “P+S”, “P+S+S” and “P+S+P” mode, a similar effect of

trailing edge can be observed in the vortex shedding process.

a) Re=30,000

In Fig.5, images of vorticity clearly describe the vortex

structure of a rhombus cylinder. For AR=0.5, the “2S” mode of

vortex shedding is observed for the rhombus cylinder. Two

single vortices shed from the cylinder per cycle of oscillation,

one by the top shear layer and another one by the bottom shear

layer. For AR=1.0, the wake is in a “P+S” mode, consisting of a

pair of vortices on one side, and a single vortex on the other

side per oscillation cycle. “2P” mode is observed for AR=1.5,

consisting of two pairs of vortices per oscillation cycle.

(a)

(b)

(c)

Fig. 5 Vortex structures for different rhombus cylinders at

Re=30,000. (a) AR=0.5, (b) AR=1.0, (c) AR=1.5.

b) Re= 60,000

The wake is still in “2S” mode for AR=0.5. But for

AR=1.0, two different modes are observed. For a period of

oscillation at a relatively low amplitude, the mode is “P+S”.

However, this mode is not stable. Fig.6 (b) shows that a

“P+S+S” mode is observed. During the downstroke of the first

cycle, the top shear layer forms a pair of vortices shed from the

trailing edge at first, then another single vortex shed from the

top shear layer. This change in wake mode corresponding to a

peak amplitude which can be seen in the displacement curve.

For AR=1.5, the wake modes are “2P” and “P+S+P”. One more

“S” shed from the shear layer per oscillation cycle, which lead

to development from “2P” to “P+S+P”. This development in the

wake pattern also generates a sudden change in the

displacement of the cylinder.


(a)

(b)

(c)


Re=60,000. (a) AR=0.5, (b) AR=1.0, (c) AR=1.5.

It should be mentioned that the wake mode (AR=1.0 and

1.5) becomes very unstable when Re>60,000. It will transform

between several different modes. The unstable wake pattern

creates chaos in the wake of a cylinder.

c) Re= 90,000

As shown in Fig.7, when Re=90,000, the wake mode of

the three rhombus cylinders is approximately the same with that

of Re=60,000. But the entrainment between the shedding

vortices are much more stronger.

d) Re= 120,000

Fig.8 (a) shows that, even when Re reaches 120,000, the

wake mode is still typically “2S” for AR=0.5. In comparison,

the wake modes of other two rhombus cylinders are more

complex. As for AR=1.0 and AR=1.5, three different modes can

be seen in the wake, including “2P”, “P+S” and “P+S+S”. The

entrainment and merging between the shedding vortices are so

strong that it’s more difficult to identify the different modes.

In order to evaluate the efficacy of energy transfer, the

mechanical power output of each oscillating rhombus cylinder

is obtained in this section. The mechanical energy transfer P between water and a transversely oscillating cylinder is

calculated on the base of a mathematical model, which was

developed by Bernitsas et al. (16). In this model, the theoretical

power output P is proportional to the fourth power of major

harmonic frequency ( oscf ) and the square of root-mean-square

amplitude ( rmsA ).

(a)

(b)

(c)


Re=90,000. (a) AR=0.5, (b) AR=1.0, (c) AR=1.5.

Fig.9 shows the mechanical power output dependence with

the AR and Reynolds number. The mechanical power output

will grow with increasing Re for each rhombus cylinder. When

Re<70,000, the power is all below 0.5-watt. But when

Re>70,000, significant increasing of P is observed. Especially

for cases of AR=1.0 and AR=1.5. The output P increases

sharply to a relatively high value. This is due to the rapidly

increases in oscillation frequency of these two cylinders, which

can be seen in Fig. 4. It should be noted that the output of

AR=1.0 is always higher than the other two cylinders when

Re>60,000 and the Re range of high output is much wider. The

maximum P reaches 5.02-watt at Re=120,000, around 50%

greater than the maximum output of AR=1.5. On the base of

analysis and comparison, the rhombus cylinder with AR=1.0 is


more suitable for harvesting energy from fluid. In other words,

it has a greater potential to drive a generator among these three

rhombus cylinders.

(a)

(b)

(c)

Fig.8. Vortex structures for different rhombus cylinders at

Re=120,000. (a) AR=0.5, (b) AR=1.0, (c) AR=1.5.

Fig. 9 Mechanical power output P (in watts).

CONCLUSIONS In this study, two-dimensional unsteady turbulent flow

over a rhombus cylinder with different axis ratio (AR) was

investigated numerically. The rhombus cylinder is constrained

to oscillate in the transverse direction. Numerical simulations

were performed for high Reynolds numbers ranging from

10,000 to 120,000 in three different axis ratios, AR=0.5, 1.0,

and 1.5. The influences of the AR on the amplitude and

frequency ratio, wake structure, and theoretical mechanical

power are obtained: 1) Both the peak amplitude ratio and root-mean-square

amplitude ratio are functions of the Reynolds numbers. As

the AR increases, higher peak amplitudes can be achieved

over a significant wide band of Re. For AR=0.5, three

branches have been observed, including the “initial

branch”, the “upper branch” and the “lower branch”.

However, the branches can not be clearly observed in the

amplitude response of the other two cylinders.

2) The frequency-ratio results show that a narrow lock-in

area is observed for AR=0.5 and AR=1.5 when

30,000<Re<50,000, but the frequency ratio of AR=1.0

monotonically increases at a nearly constant slope in the

whole Re range. It should be noted that the frequency ratio

curve of AR=1.0 keeps above the other two curves when

Re>50,000 and the maximum value reaches up to 4.1

finally.

3) The most striking feature of rhombus cylinders is the sharp

trailing edge can have a significant impact on the wake

response, and the vortex pattern becomes more complex

when the AR increases. The vortex shedding mode is

always 2S mode in the whole Re range for AR=0.5.

However, the wake patterns become diverse with the

increasing of Re for AR=1.0 and 1.5. When Re>60,000, it

will transform between several different modes.

4) The mechanical power output will grow with increasing Re

for each rhombus cylinder. For AR=1.0, the output is

always higher than the other two cylinders when

Re>60,000 and the Re range of high output is also much

wider. Thus, it can be concluded that the cylinder with

AR=1.0 has a greater potential to drive a generator among

these three rhombus cylinders.

5) The results of this paper are only limited to three different

axis ratios, the case of AR=1.0 may not be the most

appropriate rhombus cylinder for energy harvesting.

Hence, much more work must be done to fully analyze the

vibration response of rhombus with plenty of different axis

ratios. Moreover, passive control can also be used to

improve the disorder and chaos in the wake of a rhombus

cylinder, such as rounding or chamfering the corners of a

rhombus cylinder.

ACKNOWLEDGMENTS This work was supported by the Specialized Research

Fund for the Doctoral Program of Higher Education of China

(Grant No. 20120191130003).


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effects of axis ratio on the vortex-induced vibration and

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