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Research ArticleStudy on Fluid-Induced Vibration Power Harvesting ofSquare Columns under Different Attack Angles

Meng Zhang,1 Guifeng Zhao,1 and Junlei Wang2

1School of Civil Engineering, Zhengzhou University, Zhengzhou 450001, China2School of Chemical Engineering and Energy, Zhengzhou University, Zhengzhou 450001, China

Correspondence should be addressed to Junlei Wang; [email protected]

Received 4 April 2017; Revised 16 June 2017; Accepted 11 July 2017; Published 10 August 2017

Academic Editor: Micol Todesco

Copyright © 2017 Meng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A model of the flow-vibration-electrical circuit multiphysical coupling system for solving square column vortex-induced vibrationpiezoelectric energy harvesting (VIVPEH) is proposed in this paper. The quasi steady state theory is adopted to describe the fluidsolid coupling process of vortex-induced vibration based on the finite volume method coupled Gauss equation. The vibrationalresponse and the quasi steady state form of the output voltage are solved by means of the matrix coefficient method and interactivecomputing. The results show that attack angles play an important role in the performance of square column VIVPEH, of which𝛼 = 45∘ is a relatively ideal attack angle of square column VIVPEH.

1. Introduction

Recently, the development and utilization of new energysources have become a research hotspot, among which thestudy of capturing energy from environment has receivedmuch more attention. One of the most important ways ofenvironmental energy harvesting is capturing energy fromfluid, which can be divided into two kinds, wind energy andwater energy. Most of the traditional wind power and hydro-electric power facilities use the rotating turbine device to har-vest energy with large volume device and low energy density.Themicro energy technology, which can extract energy fromenvironment and convert it into electric energy [1–4], hasthe features of functional continuity, small volume, and highenergy density. In the late 1990s, the vibration piezoelectricenergy harvesting technology has beenwidely used to harvestenvironmental flow energy and convert it into vibrationenergy [5–9], which is a kind ofmicro energy technologywithcontinuous and nonconsuming energy supply. Therefore, itis an effective method for the microminiaturization of flowinduced vibration energy harvesting device.

In fluid dynamics, there is a potential physical phe-nomenon that can be used for energy harvesting called

vortex-induced vibration. Vortex-induced vibration is thatwhen the fluid flows through the bluff body, the formationand periodic shedding of the vortexwill cause the vibration ofthe bluff body. Once the vibration intensity reaches a certainlevel, the flow field shedding will be locked, which results inlarge vibration energy. In other words, vortex-induced vibra-tion is a kind of periodic, steady, or unsteady fluid structureinteraction phenomenon, which has the characteristics ofcontinuity and easy excitation [10, 11].

It is a challenging work to solve the problem of vortex-induced vibration energy harvesting. The key problem hereis how to transform the flow energy into vibration energyefficiently. In recent years, many meaningful research workshave been carried out on using vortex-induced vibration tocollect ocean energy and wind energy. Among them, theenergy conversion of circular bluff body piezoelectric vortex-induced vibration is mostly concerned. Allen and Smits [12]have studied the theory of energy harvesting of piezoelectricmaterials and designed an “eel” energy harvesting modelwhich can be used to harvest the fluid kinetic energy inthe water tank. On the basis of film theory, the “eel” modeldevice can also be used to harvest the vortex shedding energyof “lock-in” phenomenon. Taylor et al. [13] used the “eel”

HindawiGeofluidsVolume 2017, Article ID 6439401, 18 pageshttps://doi.org/10.1155/2017/6439401

https://doi.org/10.1155/2017/6439401

2 Geofluids

device of a PVDF polymer to harvest marine energy, whichwas placed in a water tank with the length of 241.3mm,the width of 76.2mm, and the thickness of 150 𝜇m. Theresearch shows that when the flapping frequency of PVDFcloses to the vortex shedding frequency, the energy collectionperformance will be improved; the maximum voltage 3Vappears in the water flow velocity of 0.5m/s. In MarineRenewable Energy Laboratory (MRELab) at the Universityof Michigan, Bernitsas et al. [14, 15] have presented a vortex-induced vibration for aquatic clean energy (VIVACE) toutilize the VIV phenomenon to generate power. The latterstudies have been conducted in support of model tests forVIVACE converter, which harnesses hydrokinetic energyenhancing flow induced motions (FIM) and particularlyVIV and various forms of galloping. Lee and Bernitsas[16, 17] built a device/system Vck to replace the physicaldamper/springs of the VIVACE with virtual elements. Thetesting was performed in the Low Turbulence Free SurfaceWater Channel of theUniversity ofMichigan at 40000 < Re

Geofluids 3

R

U

PZT

(a) Side view of the system

R K C

FyFL

F

F

M

L0

U

D

x

(b) Mass-Spring-Damping-Resistanceload system

Figure 1: Physical model of square column energy harvesting system.

system, 𝐶 is the damping of the system, and 𝑅 is the externalresistance load.

3. Mathematical Model of Energy Harvesting

To analyze the coupling process of three different fields,external flow field, vortex-induced vibration, and circuit, thispaper uses the Navier-Stokes equation to describe the vortex-induced vibration, uses a linear second-order differentialequation to describe the vortex-induced vibration of thesingle degree of freedom 𝑀-𝐶-𝐾 (mass spring damping)system, and finally uses coupled Gauss law and vibrationequations to describe the electromechanical coupling sys-tem.

3.1. Fluid Solid Coupling Model. The external flow fieldis calculated by the continuity equation and the Navier-Stokes equation. Flow simulations presented in this paperare produced by open source CFD tool OpenFOAM, whichis composed of C++ libraries solving continuum mechanicsproblems with a finite volume discretization method. Sup-pose the external flow field is 2D and unsteady. The time-dependent viscous flow solutions can be obtained by numeri-cal approximation of the incompressible unsteady Reynolds-Averaged Navier-Stokes (URANS) equations in conjunctionwith the one-equation Spalart-Allmaras (S-A) turbulencemodel [33], where a second-order Gauss integration schemewith a linear interpolation is used in the governing equa-tions for the divergence, gradient, and Laplacian terms. Fortime integration, second-order backwards Euler method isemployed. The numerical discretization scheme has second-order accuracy in space and time. A pressure implicit withsplitting of operators (PISO) algorithm is used for solvingmomentumand continuity equations together in a segregatedway. The equations of motion for the square column are

solved using a second-order mixed implicit and explicittime integration scheme. The basic equations of URANSare

𝜕𝑈𝑖𝜕𝑥𝑖 = 0,𝜕𝑈𝑖𝜕𝑡 + 𝑈𝑗

𝜕𝑈𝑖𝜕𝑥𝑗 = −1𝜌𝜕𝑝𝜕𝑥𝑖 +

𝜕𝜕𝑥𝑗 (2]𝑆𝑖𝑗 − 𝑢𝑗𝑢𝑖) ,(1)

where 𝑝 is pressure, 𝜌 is fluid density, ] is dynamic viscosity,𝑈𝑖 is the mean flow velocity vector, and 𝑆𝑖𝑗 is the strain ratetensor,

𝑆𝑖𝑗 = 12 (𝜕𝑈𝑖𝜕𝑥𝑗 +

𝜕𝑈𝑗𝜕𝑥𝑖 ) . (2)To solve the URANS equations for mean flow properties

and potential turbulence flow, the Boussinesq eddy-viscosityapproximation is adopted here, which relates to the Reynoldsstress and the velocity gradient. The quantity 𝜌𝑢𝑗𝑢𝑖 is theReynolds stress tensor and can be modeled as 𝜌𝑢𝑗𝑢𝑖 = 2𝜇𝑡𝑆𝑖𝑗,where 𝜇𝑡 is the turbulence eddy viscosity.

The S-A model is widely used for turbulence closure.The eddy-viscosity coefficient can be calculated from thefollowing transport equation:

𝜕]̃𝜕𝑡 + 𝑢𝑗 𝜕]̃𝜕𝜒𝑗 = 𝑐𝑏1�̃�]̃ − 𝑐𝑤1𝑓𝑤 (]̃𝑑)2

+ 1𝜎 { 𝜕𝜕𝜒𝑗 [(] + ]̃)𝜕]̃𝜕𝜒𝑗]} + 𝑐𝑏2

𝜕]̃𝜕𝜒𝑖⋅ 𝜕]̃𝜕𝜒𝑗 .

(3)

4 Geofluids

In S-A model, the turbulence eddy viscosity 𝜇𝑡 can beobtained by

𝜇𝑡 = 𝜌]̃𝑓V1, (4)in which 𝑓V1 = 𝜒3/(𝜒3 + 𝑐3]1), and 𝜒 ≡ ]̃/], where 𝜒 is anintermediate value. ]̃ is working variable of the turbulencemodel and depends on the transport equation (3).

The details of the transport equation are in Spalart andAllmaras [33], and the trip terms 𝑓𝑡1 and 𝑓𝑡2 are switchedoff and a “trip-less” initial condition was added for solvingworking variable ]̃. This approach was successfully used inthe work of Wu et al. [21] and Ding et al. [22] for circularcylinders.

The motion equations of a single degree of freedom𝑀-𝐶-𝐾 system can be represented by the following linearsecond-order differential equation:

𝑀�̈� + 𝐶�̇� + 𝐾𝑌 = 𝐹𝑦. (5)The relationships between𝑀, 𝐾, and 𝐶 are as follows:

𝜔𝑛 = √ 𝐾𝑀,𝐶 = 2𝜉𝑀𝜔𝑛,

(6)

where 𝐹𝑦 stands for the force on unit volume of the flow field,which is perpendicular to the flow direction, Y stands for thecolumn vibration displacement, �̇� and �̈� represent the first-or second-order derivative of the column vibration displace-ment, respectively, 𝜔𝑛 is the natural circular frequency, and 𝜉is the dimensionless damping ratio.The dynamic response ofthe column can be obtained by solving the motion equationsand the fluid governing equations simultaneously.

3.2. Electromechanical Coupling Model. In order to describethe relationship between the amplitude and the voltage in thevortex-induced vibration circuit, Gauss law is adopted in thispaper. Theoretical derivations are as follows:

𝑀�̈� + 𝐶�̇� + 𝐾𝑌 − 𝜃𝑉 = 𝐹𝑦, (7)𝜃�̇� + 𝐶𝑝�̇� + 𝑉𝑅 = 0, (8)

where 𝜃 is the electromechanical coupling coefficient, 𝐶𝑝 isthe capacitance coefficient, and 𝑉 is the voltage.

The influences of the vortex-induced vibration system onthe circuit output voltage have been considered in (7) and (8),respectively. At the same time, the negative feedback effect ofthe circuit on the vibration system is also taken into account;that is to say, the influence of electromechanical coupling isconsidered. Combined with the flow field calculation results,we can carry out the flow-mechanical-electrical couplinganalysis.

In order to solve the damping and natural frequency ofthe system, we use the matrix method to calculate the two-order nonhomogeneous ordinary differential equation (7).The homogeneous equation of (7) is as follows:

𝑀�̈� + 𝐶�̇� + 𝐾𝑌 − 𝜃𝑉 = 0. (9)

Let 𝑋1 = 𝑌, 𝑋2 = �̇�, and 𝑋3 = 𝑉; substitute (6) into (8) and(9); we can get

�̇�1 = 𝑋2�̇�2 = −𝜔𝑛2𝑋1 − 2𝜉𝜔𝑛𝑋2 + 𝜃𝑀𝑋3�̇�3 = − 𝜃𝐶𝑝𝑋2 −

𝑋3𝑅𝐶𝑝 .(10)

The above equations can be expressed in the followingmatrixform:

�̇� = 𝐵 (𝑅)𝑋, (11)where

𝑋 = [𝑋1, 𝑋2, 𝑋3]𝑇 ,

𝐵 (𝑅) =[[[[[[[

0 1 0−𝜔𝑛2 −2𝜉𝜔𝑛 𝜃𝑀0 − 𝜃𝐶𝑝 −

1𝑅𝐶𝑝

]]]]]]]. (12)

The matrix 𝐵(𝑅) has three different eigenvalues of 𝑘𝑖, ofwhich 𝑖 = 1, 2, 3. Spalart and Allmaras [33] have pointedout that the first two eigenvalues are similar to the ones ofvibration system without circuit, and yet the third eigenvalueis associated with the electromechanical coupling effect, suchas piezoelectric system affected by the foundation or theaeroelastic excitation, and is negative constant. There areconjugate relations between 𝑘1 and 𝑘2, in which the real partand the imaginary part of the conjugate solution stand forthe damping and natural frequency of the electromechanicalcoupling system, respectively. Given that 𝑘3 is negativeconstant, we consider only the real part of 𝑘1 and 𝑘2, whencomputing the trivial solution of the matrix 𝐵(𝑅).4. Quasi Steady State Model forOutput Voltage

In this section, we adopt the quasi steady state modelproposed by Barrero-Gil et al. [34] to describe the amplitudeof vortex-induced vibration and calculate the time-varyingvibration energy harvesting.When the vortex-induced vibra-tion is in the synchronization region, the vibration amplitudeof the system can be expressed as the following sine function:

𝑌 = 𝑌max sin (𝜔𝑛𝑡) , (13)where 𝑌max is the maximum column vibration displacement.

It is worth noting that the voltage time-history curve andthe vibration amplitude time-history curve are synchronous;that is to say, there is no phase difference. Substituting (13)into (11), we can get the analytical solution of the quasi steadystate voltage with MATLAB.

𝑉 (𝑡) = 𝜃𝜔𝑛𝑅𝑌max1 + 𝜔𝑛2𝑅2𝐶𝑝2 (𝑒−𝑡/𝑅𝐶𝑝 − cos (𝜔𝑛𝑡)

− 𝜔𝑛𝑅𝐶𝑝 sin (𝜔𝑛𝑡)) .(14)

Geofluids 5

Table 1: Calculation parameters for square column VIVPEH system.

Symbols and units Physical meanings Numerical values𝑀 [kg] Square column mass 0.2979𝐾 [N/m] Elastic coefficient of the system 579∼584𝐶 [N⋅s/m] Damping of the system 0.0325∼0.45𝐷squ [m] Square column side length 0.0016𝛼 [degree] Angle between the square column and the inflow velocity 0∼75𝜁 Damping ratio of the system 0.00121𝑓𝑛 [Hz] Natural frequency of the system 7.012∼7.132 (water)𝐶𝑝 [nF] Capacitance 120𝜃 Electromechanical coupling coefficient 1.55 × 10−3𝜇 [Pa⋅s] Dynamic viscosity 0.0011379] [m2/s] Kinematic viscosity 1.139 × 10−6 (water)𝜌 [kg/m3] Density 999.1026 (water)

The corresponding output power can be obtained by thefollowing equation:

𝑃 (𝑡) = 𝑉2 (𝑡)𝑅 . (15)The above mathematical expressions include the solving

process of fluid solid coupling and electromechanical cou-pling.

First of all, we can get the flow field pressures 𝑃 and 𝐹𝑦by solving the URANS equations in OpenFOAM. Secondly,we can obtain the column vibration displacement 𝑌, thedamping 𝐶, and natural circular frequency 𝜔𝑛, by solving (5)to (11). It should be noted that the motion of the columnis influenced by the pressure of the flow field. At the sametime, the vibration of the column gives feedback to the flowfield and causes the change of flow field distribution. Thus,the fluid solid coupling problem can be solved by interactivecomputing. Finally, the time-history curve of output voltageand output power can be obtained by (14) and (15). Theabove is the whole computing process of flow-vibration-electromechanical coupling system.

5. Numerical Results

5.1. Analysis Parameters and Cases. To carry out the numer-ical calculation of vortex-induced vibration, this paper pro-vides the parameters of the vibration energy harvestingsystem, as shown in Table 1, in which𝐷squ is the side length ofthe square column, 𝐷Nor stands for the dimensionless char-acteristic length of the square column, and the relationshipbetween𝐷Nor and𝐷squ is given in

𝐷Nor = 𝐷squ (sin𝛼 + cos𝛼) . (16)Based on the parameters of Table 1 and formula (16), we cancalculate the numerical values of 𝐷Nor as 0.0016 under 0∘attack angle, 0.00196 under 15∘ attack angle, 0.00219 under 30∘attack angle, 0.00226 under 45∘ attack angle, 0.00219 under60∘ attack angle, and 0.00196 under 75∘ attack angle.

The computational grid and boundary conditions ofsquare column vortex-induced vibration under different

attack angles are given in Figure 2, where the computationaldomain is 20 × 20D, and the entire domain includes fiveboundaries: velocity inlet, velocity outlet, top, bottom, anda column wall. The inlet velocity is considered as uniformand constant velocity. For outlet boundary, a zero gradientcondition is specified for velocity. The top and bottom con-dition are defined as a wall boundary. In present numericalstudy, a moving wall boundary condition is applied forthe square column when the column is in VIV. The two-dimensional, structured grids were generated with the helpof “Gambit” software. The grid domain size is 20 × 20D. Thesquare column was set in the center in the domain to ensurethat the results of the numerical model are accurate. Theconditions at the outlet are close to the assumed conditions.The computational domain in the vicinity of each cylinder is a3 × 3D square where the grid density for the near-wall regionis enhanced to solve for high resolution in flow properties.

When the external resistance value is 𝑅 = 1 × 106Ω, thespring stiffness value of the square column VIVPEH systemis 𝐾squ = 580N/m, and the corresponding damping value is𝐶squ = 0.2Ns/m. Taking the range values of the flow velocityas 0.03927m/s to 0.08975m/s, we can calculate the numericalvalues of 𝑈𝑟squ under different attack angles, as shown inTable 2, in which 𝑈𝑟squ is the reduction velocity, defined asfollows:

𝑈𝑟squ = 𝑈𝑓𝑛𝐷squ , (17)where 𝑓𝑛 = 𝜔𝑛/2𝜋.

In this paper, the matrix method is used to evaluate theinfluence of the external resistance load on the damping andnatural frequency of the electromechanical coupling systemby MATLAB. Then, the damping and natural frequencyvalues can be used for initial conditions in OpenFOAMto compute the vibration amplitude of the vortex-inducedvibration system under different Reynolds numbers (94

6 Geofluids

(a) 𝛼 = 0∘ (b) 𝛼 = 15∘ (c) 𝛼 = 30∘

(d) 𝛼 = 45∘ (e) 𝛼 = 60∘ (f) 𝛼 = 75∘

Square column

Top

y

x

OutletInlet

Bottom

(g)

Figure 2: Computational grid and boundary conditions.

5.2. System Damping and Natural Frequency Characteristics.The damping and natural frequency of the electromechanicalcoupling system can be obtained by MATLAB, and the realand imaginary parts of the circuit conjugate solutions areshown in Figure 3.

According to Figure 3, we can see that the total dampingof the system is small when the resistance load is small.In the case of 𝑅 < 1 × 105Ω, the system total dampingis increased with the resistance load increasing. Once the

resistance load 𝑅 reaches 1 × 105Ω, the system total dampingreaches the maximum value. It is noteworthy that the totaldamping of the system decreases instead of increasing whenthe resistance load keeps increasing; that is, 𝑅 > 1 × 105Ω.For the natural circular frequency, when 𝑅 < 3 × 104Ω, thefrequency value remains at 44 rad/s; when 𝑅 > 2 × 106Ω, thefrequency value approximately remains at 50 rad/s. Generally,the natural circular frequency value of the system is relativelystable, which is kept in the range of 44–50 rad/s.

Geofluids 7

Table 2: Run case of square column in OpenFOAM.

𝑈 (m/s) 𝑈𝑟squ𝛼 = 0∘ 𝛼 = 15∘ (𝛼 = 75∘) 𝛼 = 30∘ (𝛼 = 60∘) 𝛼 = 45∘0.03927 3.5 2.85784 2.56223 2.475250.04488 4 3.26611 2.92826 2.828850.05049 4.5 3.67437 3.29429 3.182460.0561 5 4.08263 3.66032 3.536070.05834 5.2 4.24594 3.80673 3.677510.06059 5.4 4.40924 3.95315 3.818950.06283 5.6 4.57255 4.09956 3.96040.06732 6 4.89916 4.39239 4.243280.07293 6.5 5.30742 4.75842 4.596890.07854 7 5.71569 5.12445 4.95050.08415 7.5 6.12395 5.49048 5.30410.08976 8 6.53221 5.85652 5.657710.09537 8.5 6.94048 6.22255 6.01132

Real Imaginary

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Real

103 104 105 106 107102

R (ohms)

44

45

46

47

48

49

50

Imag

inar

y

Figure 3: Real and imaginary parts of the circuit conjugate solu-tions.

5.3. Vibration Characteristics of Square Column VIVPEHunder Different Attack Angles. In the following numericalsimulations, we take the resistance load 𝑅 = 1 × 106Ωand compute the amplitude response of the square columnVIVPEH under different attack angles and different flowvelocities, as shown in Figure 4. Note here that the naturalfrequency of square column is 𝑓𝑛 = 6.98Hz, when theresistance load is 𝑅 = 1 × 106Ω. The dimensionless vibrationamplitude 𝑌squ/𝐷squ is adopted to indicate the vibrationresponse of the square column. The maximum value of 𝑌squcan be obtained by means of averaging the peak value of atleast 60 displacement time-history response curves. In thispaper, the vibration amplitude of the smooth circular columnis provided to compare with that of the square column.

As we can see in Figure 4, different attack angles obvi-ously have an effect on the peak vibration amplitude andlock-in region. Moreover, we can observe the “presynchro-nization,” “synchronization,” and “postsynchronization” ofvortex-induced vibration curves of square column VIVPEHunder different attack angles.

0.0

0.1

0.2

0.3

0.4

5 83 74 6UrSqu

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

/DSq

uY

Squ

Figure 4: Dimensionless vibration amplitude of square columnVIVPEH under different attack angles and different flow velocities.

5.3.1. 𝛼 = 0∘. The lock-in region is from 𝑈𝑟squ = 6.3 to theend of 𝑈𝑟squ = 6.7, and the maximum amplitude is 𝑌squmax/𝐷squ = 0.05, which appeared at 𝑈𝑟squ = 6.5, as shown inFigure 4. In order to see more clearly about the vibrationamplitude 𝑌squ/𝐷squ, some displacement time-history curvesand FFT analysis results for 𝛼 = 0∘are given in Figure 5.It can be seen that when 𝑈𝑟squ is small, the amplitude oftime-history curve presents a stable sinusoidal curve, and themaximum amplitude is so small that it can be ignored, whichmeans that there is almost no vortex-induced vibration in thesystem.With the increases of𝑈𝑟squ, the systementers “presyn-chronization” phase and the vibration amplitude 𝑌squ/𝐷squgradually increases. When the vortex shedding frequencyapproaches the natural frequency of square columnVIVPEH,the oscillation amplitude of the square column VIVPEH

8 Geofluids

−0.01

0.00

0.01

6 8 10 124Time (s)

/DSq

uY

Squ

(a) Nondimensional displacement for𝑈𝑟squ = 6.0

0

200

400

600

800

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)

(b) FFT analysis for𝑈𝑟squ = 6.0

−0.06

−0.04

−0.02

0.00/DSq

uY

Squ

0.02

0.04

0.06

18 21 24 27 3015Time (s)

(c) Nondimensional displacement for𝑈𝑟squ = 6.5

0

1000

2000

3000M

agni

tude

4 5 6 7 8 9 103Frequency (Hz)

(d) FFT analysis for𝑈𝑟squ = 6.5

Figure 5: Displacement time-history and FFT analysis results for 𝛼 = 0∘.

increases significantly and the system enters synchronousphase, which means the phenomenon of “lock-in” occurs.Under synchronous phase, the vortex shedding frequency iskept constant and the amplitude of the system will remain ata higher value when the flow velocity 𝑈𝑟squ is increased from𝑈𝑟squ = 6.3 to the end of 𝑈𝑟squ = 6.7.5.3.2. 𝛼 = 15∘. It can be seen from Figure 2(b) that thereis a corner at the upper windward of the square column,which can obviously affect the flow field. As is shown inFigure 5, the maximum amplitude decreases and the lock-in region is narrow (from 𝑈𝑟squ = 5.4 to 𝑈𝑟squ = 5.7). Themaximum amplitude is𝑌squmax/𝐷squ = 0.148, which appearedat𝑈𝑟squ = 5.7.The displacement time-history curves and FFTanalysis results for 𝛼 = 15∘ are given in Figure 6. Similar withthe results of 𝛼 = 0∘, the displacement time-history curvecan also be divided into the following four stages: “unsyn-chronization,” “presynchronization,” “synchronization,” and“postsynchronization.”

5.3.3. 𝛼 = 30∘. The results in Figure 4 show that the max-imum amplitude of the square column is obviously increased

with a wider range of lock-in region (from 𝑈𝑟squ = 4.5to 𝑈𝑟squ = 5.7); the maximum amplitude 𝑌squmax/𝐷squreaches 0.41. Similarly, some of the displacement time-historycurves and FFT analysis results for 𝛼 = 30∘ are given inFigure 7.The results in Figure 7(a) show that the whole time-history curve appears as a parabolic shape and the growthrate decreased gradually to a stable level at late stage, whichindicates the coupling of the vortex shedding frequency andnatural frequency; that is, the lock-in phenomenon occurs.It can also be seen that there is a “beat” phenomenon inthe amplitude curves, as shown in Figure 7(a). There arethree peaks in the frequency spectrum curve, as shown inFigure 7(b), which indicates that three harmonics occur inthe system. The explaining of the above phenomena is asfollows. In the process of vortex-induced vibration of thesquare column, the vortex shedding frequency couples withthe natural frequency when the flow velocity 𝑈𝑟squ reaches acertain value.The flow field near the square column surface isstrongly disturbed because of the corner point of the squarecolumn, which results in the superposition of three vibrationfrequencies. This is because the unbalance of the upper andlower aerodynamic force is caused by the obvious asymmetry

Geofluids 9

−0.03

−0.02

−0.01

0.00/DSq

uY

Squ

0.01

0.02

0.03

18 21 24 2715Time (s)


0

200

400

600

800

1000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


−0.15

−0.10

−0.05

0.00/DSq

uY

Squ

0.05

0.10

0.15

18 21 2415Time (s)


0

2000

4000

6000

8000

10000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)



under attack angle 𝛼 = 30∘. When the flow velocity is about𝑈𝑟squ = 5.49, as shown in Figures 7(c) and 7(d), the amplitudecurve appears as a complete sine curve without noise. Themaximum amplitude 𝑌squmax/𝐷squ is increased to about 0.41and the vibration frequency is stabilized at 6.98Hz, which canbe regarded as the best working condition of square columnVIVPEH.

5.3.4. 𝛼 = 45∘. As is shown in Figure 4, the displacementtime-history curve can also be divided into the follow-ing four stages: “unsynchronization,” “presynchronization,”“synchronization,” and “postsynchronization.” The maxi-mum amplitude is 𝑌squmax/𝐷squ = 0.28 and the lock-inregion is from 𝑈𝑟squ = 4.2 to the end of 𝑈𝑟squ = 5.4,which appears earlier than that of other attack angles. Thedisplacement time-history curves and FFT analysis resultsfor 𝛼 = 45∘ are given in Figure 8. It can be seen that thereis no severe aerodynamic disturbance, which shows that thevortex-induced vibration response of symmetric bluff body isrelatively stable.

5.3.5. 𝛼 = 60∘. The displacement time-history curves andFFT analysis results for 𝛼 = 60∘ are given in Figure 9, which

are similar in shape to the one for 𝛼 = 30∘ with only slightdifferences in values. According to the displacement time-history curves, as shown in Figures 7 and 9, it can be seen thatthe amplitude results are almost exactly the same as the onefor 𝛼 = 30∘ in the lock-in region. The same conclusion canalso be obtained from the spectral analysis results; that is, thephenomena of harmonic and noise are similar.This is becausethe spring force of the system and the gravity of the columnprovide a balance in the flowfield. So it can be considered thatthe above two cases of 𝛼 = 30∘ and 𝛼 = 60∘ are symmetric.5.3.6.𝛼 = 75∘. Theresults in Figure 10 show the displacementtime-history curves and FFT analysis results for 𝛼 = 75∘. Itis easy to see that both the displacement time-history curvesand the spectral analysis results are similar to those of 𝛼 = 15∘case, which indicates that the cases of 𝛼 = 15∘ and 𝛼 = 75∘are also symmetric.

In order to verify the above conclusions, the StrouhalNumbers for the following four cases, 𝛼 = 15∘, 𝛼 = 30∘,𝛼 = 60∘, and 𝛼 = 75∘, are compared with each other, as shownin Table 3. It can be seen that the Strouhal Number results forthe case of 𝛼 = 15∘ (𝛼 = 30∘) are almost equal to those for

10 Geofluids

−0.2

−0.1

0.0/DSq

uY

Squ

0.1

0.2

3 6 9 12 15 18 21 24 27 300Time (s)


0

2000

4000

6000

8000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

/DSq

uY

Squ

3 6 9 12 15 18 21 24 27 300Time (s)


0

10000

20000

30000

40000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)



Table 3: Comparison of Strouhal Number with different attackangle.

U (m/s) Strouhal Number15∘ 75∘ 30∘ 60∘

0.05049 0.1682 0.1682 0.1832 0.18310.0561 0.1714 0.1714 0.1873 0.18720.06278 0.1743 0.1742 0.1896 0.18960.07293 0.1746 0.1746 0.1922 0.1922

the case of 𝛼 = 75∘ (𝛼 = 60∘), which indicates that the aboveanalysis is correct.

To show the difference more clearly of the lock-in regionof square column VIVPEH under different attack angles, wechoose the dimensionless frequency 𝑓squ/𝑓𝑛squ to indicatethe frequency characteristics of square column VIVPEH,where 𝑓squ is the vortex shedding frequency of the system,which can be obtained by Fast Fourier Transform (FFT)of the displacement time-history curve, shown in Figures5–10; 𝑓𝑛squ is the natural frequency of the square column.The results in Figure 11 show the different lock-in regionsof square column VIVPEH under different attack angles.

For the case of 𝛼 = 0∘, the range of values for the lock-inregion is 6.5 to 7.0 (6.5 ≤ 𝑈𝑟squ ≤ 7.0); the correspondingbandwidth value is 0.5. For the case of 𝛼 = 15∘ and 𝛼 =75∘, the range of values for the lock-in region is 5.4 to 5.7(5.4 ≤ 𝑈𝑟squ ≤ 5.7); the corresponding bandwidth valueis just 0.3. For the case of 𝛼 = 30∘ and 𝛼 = 60∘, therange of values for the lock-in region is 4.6 to 5.5 (4.6 ≤𝑈𝑟squ ≤ 5.5), the corresponding bandwidth value is 0.9,which is significantly larger than that of the above two cases.When the attack angle is 𝛼 = 45∘, the range of values forthe lock-in region is 4.2 to 5.4 (4.2 ≤ 𝑈𝑟squ ≤ 5.4); thecorresponding bandwidth value increases to 1.2. In addition,the three different branch types of square column VIVPEHcan be observed in Figure 11, such as “presynchronization,”“synchronization,” and “postsynchronization.”

Based on the above analysis results, we believe that thecalculation parameters and response parameters for the caseof 𝛼 = 15∘ and 𝛼 = 30∘ are about the same as those for thecase of 𝛼 = 75∘ and 𝛼 = 60∘.5.4. Phase Angle Analysis of Square Column VIVPEH underDifferent Attack Angles. The analysis results of the initialphase angle and the synchronous phase angle of square

Geofluids 11

−0.006

−0.004

−0.002

0.000

0.002

0.004

0.006/D

Squ

YSq

u

1815 21 2724 3330Time (s)


0

100

200

300

400

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

/DSq

uY

Squ

18 20 2216Time (s)


0

5000

10000

15000

20000

25000

30000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)



column VIVPEH under different attack angles are given inFigures 12–15. It can be seen that the vibration is stable inthe initial stage with a single amplitude and frequency, whichis completely determined by the vortex shedding frequency.Therefore, the displacement time-history curve is almostsynchronous with the lift coefficient curve without phasedelay. This is because the amplitude is small, so that there isalmost no stagnation when the vibration amplitude reachesits maximum or minimum value. Different from the initialvibration state, the vortex shedding frequency is locked againin the synchronous state, resulting in a multiple relationshipbetween the vibration frequency and the natural frequency.That means, in synchronization region, there exists no phasedifference between the vortex shedding frequency and vibra-tion frequency. Accordingly, the displacement time-historycurve is completely synchronized with the lift coefficientcurve with some phase delay.

Figure 16 shows that the vortex-induced vibration ofsquare column VIVPEH will stagnate for some time at thewave crest or the wave trough, because of the buffering effectof spring. As is shown in Figure 16(a), the vortices of 𝑉1 and𝑉2 move forward to a distance of 𝑆1 when the wave crests ofthe two adjacent steps appear. At the moment, 𝑉1 obviously

becomes thinner and longer, while the vibration of the squarecolumn is still in the wave crest. Similarly, the vortices of𝑉3 and 𝑉4 move forward to a distance of 𝑆2 when the wavetroughs of the two adjacent steps appear. It should be pointedout that, the vibration of the square column is always in thewave crest or the wave trough, which leads to the generationof the phase angle.

5.5. Analysis of Near Wake Vortex Shedding of Square ColumnVIVPEH under Different Attack Angles. The shapes of wakevortices in synchronized state of square column VIVPEHunder different attack angles are given in Figures 17–20, inwhich 𝑇 is the vibration period with subscript representingthe value of an attack angle. The direction of the negativevorticity region is counterclockwise, which is expressed inblue; while the direction of the positive one is clockwise,which is expressed in red. It can be seen that, for the case of𝛼 = 0∘, the flow field around the square column is stable, thevibration amplitude is small, and the wake vortex structurepresents a regular 2S shape. In other words, a positive andnegative vortex pair sheds in a cycle. For 𝛼 = 15∘/𝛼 = 75∘,the position of near wake vortices shedding of square columnVIVPEHbeganmoving forward to the near columnwall with

12 Geofluids

−0.2

−0.1

0.0

0.1

0.2

/DSq

uY

Squ

5 10 15 20 25 30 35 400Time (s)


0

5000

10000

15000

20000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


−0.4

−0.2

0.0

0.2

0.4

/DSq

uY

Squ

3 6 9 12 150Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

2000

4000

6000

8000

10000

12000

14000

16000

Mag

nitu

de



increasing vibration amplitudes. Correspondingly, the wakevortex shedding array gradually changes its shape from 2Sshape to a circle. As for 𝛼 = 30∘/𝛼 = 60∘, it can be observedthat the wake vortex shedding mode changes obviously withan increasing width of the vortex array, due to the increaseof vibration amplitude and the shape of the vortex pair ischanged from a flat shape to a regular elliptical shape. Whenthe attack angle is 𝛼 = 45∘, the vibration amplitude decreases,which results in the wake vortex shedding mode changingback to the stable 2S mode.

5.6. Voltage Output and Power Output of Square ColumnVIVPEH under Different Attack Angles. Due to the influenceof different attack angles, the vortex separation point ofsquare column VIVPEH is different from that of cylinder,which results in the following analysis being more compli-cated. Considering the symmetric nature of the case 𝛼 =15∘/𝛼 = 75∘ and the case 𝛼 = 30∘/𝛼 = 60∘, we take only thecase of 𝛼 = 0∘, 𝛼 = 15∘, 𝛼 = 30∘, and 𝛼 = 45∘ in the followinganalysis of voltage output. When the resistance load is 𝑅 = 1× 106Ω, the output voltage of the system can be calculated by(14).

The purpose of this section is to investigate themaximumvalue of the output voltage and the lock-in region to select theoptimal attack angle of square column VIVPEH. The resultsof the maximum output voltage of square column VIVPEHunder different attack angles and the effective working area ofsynchronization are given in Figure 21. It can be seen that themaximum output voltage of square columnVIVPEH appearsat the case of 𝛼 = 45∘ and the corresponding value is 6.732V,while the minimum value appears at the case of 𝛼 = 15∘/𝛼 =75∘, which indicates that these two cases are not suitable forenergy harvesting. Similarly, the output voltage of the case of𝛼 = 0∘ is slightly higher than that of the case of 𝛼 = 15∘/𝛼 =75∘; however, its value is still small. When the attack angleincreases from 𝛼 = 15∘ to 𝛼 = 45∘, the output voltage ofsquare column VIVPEH system increases to its maximumvalue.The results of the working region of synchronization inFigure 21 show that the total bandwidth of the above six casesis (4.2–5.7, 5.3–5.7), and there are about 0.6 bandwidth of theregion of nonsynchronization. The maximum bandwidth is(4.2–5.4) under attack angle 𝛼 = 45∘, which is higher thanthat of all other attack angle cases. Accordingly, the outputpower of the system can be calculated by (15).

Geofluids 13

−0.2

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0.0

0.1

0.2

/DSq

uY

Squ

18 21 24 27 30 3315Time (s)


0

500

1000

1500

2000

2500

3000

Mag

nitu

de

4 5 6 7 8 9 103Frequency (Hz)


−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

/DSq

uY

Squ

18 21 24 27 3015Time (s)


4 5 6 7 8 9 103Frequency (Hz)

0

4700

9400

14100

18800M

agni

tude



0.6

0.8

1.0

1.2

85 6 743

= 0∘

= 15∘

= 30∘

= 45∘

= 60∘

= 75∘

f３ＫＯ/f

N３ＫＯ

f３ＫＯ/fN３ＫＯ = 1

UrSqu

Figure 11: Nondimensional frequency of square column VIVPEH under different attack angles.

14 Geofluids

−0.004

−0.003

−0.002

−0.001

0.000

0.001

0.002

0.003

0.004

Y/D

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

CL

15.815.4 15.6 16.015.0 15.2

Time (s)

(a) Initial phase angle

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

Y/D

15.815.4 15.6 16.015.0 15.2

Time (s)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

CL

(b) Synchronous phase angle

Figure 12: Results of phase angle in initial and synchronization state for 𝛼 = 0∘.

−0.004

−0.002

0.000

0.002

0.004

Y/D

15.2 15.4 16.015.815.615.0

Time (s)

−1.0

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1.0

CL


−0.15

−0.10

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0.00

0.05

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0.15Y/D

15.2 15.4 15.6 15.8 16.015.0

Time (s)

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

CL


Figure 13: Results of phase angle in initial and synchronization state for 𝛼 = 15∘/𝛼 = 75∘.

−0.002

0.000

0.002

Y/D

15.1 15.815.715.6 15.915.0 15.415.315.2 16.015.5

Time (s)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

CL

−0.004

0.004


−0.4

−0.3

−0.2

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0.0

0.1

0.2

0.3

0.4

Y/D

15.1 15.2 15.915.4 15.815.6 15.7 16.015.0 15.3 15.5

Time (s)

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5C

L


Figure 14: Results of phase angle in initial and synchronization state for 𝛼 = 30∘/𝛼 = 60∘.

Geofluids 15

−0.006

−0.004

−0.002

0.000

0.002

0.004

0.006Y/D

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 16.015.0

Time (s)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

CL


15.1 15.515.3 15.7 15.915.615.2 15.815.4 16.015.0

Time (s)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

CL

−0.2

−0.1

0.0

0.1

0.2

Y/D


Figure 15: Results of phase angle in initial and synchronization state for 𝛼 = 45∘.

V1

V1

V2

V2

S1

(a) 𝑌max+ (wave crest)

V3

V4

S2

V3

V4

(b) 𝑌max− (wave trough)

Figure 16: Diagram of phase angle between displacement and lift force development history.

t/T３ＫＯ0 = 0 t/T３ＫＯ0 = 0.249

t/T３ＫＯ0 = 0.502 t/T３ＫＯ0 = 0.75

Figure 17: Vortex structures in synchronization region for 𝛼 = 0∘.

The results of the maximum output power of squarecolumn VIVPEH under different attack angles are givenin Figure 22. It can also be seen that the maximum valueof the output power appears at the case of 𝛼 = 45∘ and

t/T３ＫＯ15 = 0 t/T３ＫＯ15 = 0.25

t/T３ＫＯ15 = 0.497 t/T３ＫＯ15 = 0.752

Figure 18: Vortex structures in synchronization region for 𝛼 =15∘/𝛼 = 75∘.

the corresponding value is 4.5 × 10−5W. Therefore, squarecolumn VIVPEH under attack angle 𝛼 = 45∘ is an ideal PEH,because of its higher voltage output value and larger workingbandwidth.

16 Geofluids

t/T３ＫＯ30 = 0

t/T３ＫＯ30 = 0.499

t/T３ＫＯ30 = 0.248

t/T３ＫＯ30 = 0.749

Figure 19: Vortex structures in synchronization region for 𝛼 =30∘/𝛼 = 60∘.

t/T３ＫＯ45 = 0

t/T３ＫＯ45 = 0.499

t/T３ＫＯ45 = 0.252

t/T３ＫＯ45 = 0.751

Figure 20: Vortex structures in synchronization region for 𝛼 = 45∘.

6. Conclusions

The energy harvesting features of square column VIVPEHunder different attack angles are investigated in this paperwith considering the vibration characteristics, phase charac-teristics, the near wake vortex sheddingmode, and the outputvoltage and power of the system.Themain conclusions are asfollows:

(1) Within the range of reduced velocity studied in thispaper, the vortex-induced vibration curves of squarecolumn VIVPEH under different attack angles can beobtained, which contains the phenomena of “presyn-chronization,” “synchronization,” and “postsynchro-nization.” The vortex shedding shape of square col-umn VIVPEH is dominated by 2S mode.

(2) The attack angle has significant effect on the maxi-mum value of vibration amplitude of square columnand the lock-in region. Due to the influence ofdifferent attack angles, the boundary layer separationpoint does not move backwards like the cylinder.The maximum vibration amplitude and the lock-invibration region of square column will also fluctuate.When the attack angle is equal to 45 degrees, thesynchronization region can reach a value of 1.2 timesof reduction velocity.

(3) The numerical analysis shows that the cases of 𝛼 =15∘/𝛼 = 30∘ and 𝛼 = 75∘/𝛼 = 60∘ for squarecolumnVIVPEH are symmetric, which indicates that

(6.3–6.7)(5.4–5.7)

(a–b) is the Ur rangeof the synchronization

effective area

(5.0–5.5)

(4.2–5.4)

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

2

3

4

5

6

7

Max

val

ue o

fV＃２－

３(V

)

Figure 21: Comparison of maximum value of voltage output andsynchronization effective area between VIVPEH with differentshapes.

0.0

1.0 × 10−5

2.0 × 10−5

3.0 × 10−5

4.0 × 10−5

5.0 × 10−5

= 15∘ Squ = 30∘ Squ = 45∘ Squ = 0∘ Squ

PＧ；Ｒ

)M

ax v

alue

of p

ower

gen

erat

ed (

Figure 22: Comparison of maximum value of power outputbetween VIVPEH with different shapes.

the calculated results of the case of 𝛼 = 15∘/𝛼 = 30∘are equal to the results of case of 𝛼 = 75∘/𝛼 = 60∘,respectively.

(4) The maximum value of the output power is similarto that of the output voltage, which appears at thecase of 𝛼 = 45∘ and the corresponding value is 4.5× 10−5W and 6.732V, respectively. Therefore, 𝛼 =45∘ is a relatively ideal attack angle of square columnVIVPEH.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

This research has been funded by the National NaturalScience Foundation of China (Grants nos. 51606171, 51578512,and 51108425) and the Outstanding Young Talent ResearchFund of Zhengzhou University (Grant no. 1521322004).

Geofluids 17

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