spectral features of the pressure distribution around a cylinder oscillating in still water.pdf

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Coastal Engineering Journal, Vol. 57, No. 3 (2015) 1550013 (19 pages)c© World Scientific Publishing Company and Japan Society of Civil Engineers

DOI: 10.1142/S0578563415500138

Spectral Features of the Pressure Distribution Arounda Cylinder Oscillating in Still Water

Seyed Taghi Omid Naeeni∗ and Seyede Masoome Sadaghi†

School of Civil Engineering, College of Engineering,University of Tehran, Tehran, Iran

∗[email protected]†[email protected]

Rangaswami Narayanan

Department of Civil and Structural Engineering,University of Manchester, Manchester, UK

narayanan [email protected]

Received 9 October 2013Accepted 13 July 2015

Published 25 August 2015

In this paper, the pressure field on a circular cylinder oscillating close to a plane bed instill water is investigated to assess the effects of proximity of the cylinder to the bed. Therange of Keulegan–Carpenter (KC) numbers of the oscillatory cylinder in the present studyis between 15 and 40. The flow is in the subcritical regime since the Reynolds number ofoscillating cylinder was between 9 500 and 26 000. Two different gaps between the cylinderand the plane expressed as ratios of the diameter of the cylinder are 0.1 and 1.0. Thecylinder and the plane bed are both smooth. The frequency spectra of pressure measured at36 positions around the cylinder have been obtained for the different KC numbers and thegap ratios. A wavelet analysis is used for the simultaneous time–frequency representationof the pressure fluctuations in order to identify the flow induced transitory characteristics.Such a representation shows clearly the abrupt changes in the pressure distribution thatare attributed to vortex shedding. The results show that changes in the pressure variationsare not at the same phase with the velocity fluctuations indicating the inability of pressuresto respond to velocity variations simultaneously.

Keywords: Oscillatory cylinder; pressure; spectral analysis; wavelet.

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http://dx.doi.org/10.1142/S0578563415500138

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S. T. O. Naeeni, S. M. Sadaghi & R. Narayanan

1. Introduction

Submarine pipelines play a very important role in the economical and safe trans-portation of oil, gas, petroleum products and even fresh water. Communicationcables are housed in a pipeline to protect from the sea environment. Sub-sea pipelinesare simply cylinders placed very close to the sea bed. Flow around circular cylindersis one of the classic and challenging topics in fluid mechanics which has attractedattention over a considerable number of years. The characteristics of flow aroundcircular cylinders are becoming clearer due to intensive studies of various investiga-tors over the years but there are still many aspects to be studied which would leadto further understanding of this complex yet fascinating phenomenon of fluid flow.

The growth and shedding of vortices from a submerged body affect the velocityfield and consequently the pressure field. The action of this pressure on the surfaceof the pipeline in turn imposes time-varying forces on the pipe. The flow imposes afluctuating component due to the alternate vortex shedding exhibiting periodicity.On the other hand, the vortex shedding in oscillatory flows is complicated by thefact that the near cylinder flow field has to rebuild itself twice per complete cycle asthe cylinder reverses its direction. The cyclic fluid motion leads to the formation ofstrong and usually asymmetric vortices which produce asymmetric flow as they areswept back past the cylinder in the return half of the flow cycle. This complicatedflow field affects the pressure distribution on the cylinder which is responsible forthe time-varying in-line and lift forces. Over the years, the vortex shedding processand its relation to the in-line and lift forces have been extensively investigated byforce measurements and flow visualization.

A few investigators have measured pressure distributions induced by periodicflow around the cylinder [Isaacson, 1974; Matten et al., 1979; Sumer et al., 1991;Dadman and Narayanan, 1995; Naeeni and Narayanan, 2005].

Isaacson [1974] studied forces on an oscillating circular cylinder using a singlepressure tapping which could be rotated to measure pressure distribution aroundthe cylinder. He reported that the signals of the flow did not vary from cycle to cycleover the entire recording period. Visualization of the flow carried out by Isaacsonunder various conditions indicated that the flow was indeed reasonably periodic forsmall and moderate Keulegan–Carpenter (KC) number, whereas for higher valuesof KC number, irregularities were observed. KC is defined as UmT/D where Um isthe maximum velocity, T is the wave period and D is the diameter of the cylinder.It has been shown that for conditions in which diffraction effects are negligible, lifton the cylinder in the wave flow is strongly nearly similar to that in the oscillatingcase.

Matten et al. [1979] obtained pressure distribution around a horizontal circularcylinder oscillating in still water. In order to obtain an essentially two-dimensionalflow, they fitted plates at the two ends of the cylinder, thus, eliminating the endeffects on the flow around the cylinder. Matten et al. were able to identify vortex

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Spectral Features of the Pressure Distribution Around a Cylinder

shedding from the pressure distribution measured for Re = 2.7 × 105 and KC = 18.Forces were obtained by integration of the pressure distribution. They found thein-line forces and transverse forces in their frequencies and amplitudes which aresomewhat similar to those observed by Sarpkaya [1976]. They also reported thatthe absence of the end plates had a substantial effect on the resultant force. Theyconcluded that force coefficients obtained from two-dimensional flow experimentscould not be directly applied to three-dimensional flows.

Sarpkaya and Wilson [1984] made differential pressure measurements on smoothand rough isolated circular cylinders in harmonically oscillating flow in a U-tube.They measured pressure distribution by rotating the cylinder. They also measuredthe velocity of flow simultaneously for use as a reference signal for the pressure. Itwas found that the growth and motion of vortices affected the pressure distributionin a profound and yet undetectable way.

Sumer et al. [1991], measured instantaneous pressure distributions around anoscillatory cylinder over a plane bed for various values of gap ratios between 0 and2, KC being between 4 and 65. Both smooth and rough cylinders were considered.The gap ratio is G/D where G is the gap distance between the bottom of the cylinderand the flume’s bed and D is the diameter of the cylinder. In their experiments theyused 12 pressure transducers mounted on the surface of the cylinder. They reportedthat the pressure distributions were repeatable over the cycles. Hydrodynamic forceswere determined by integration of pressure distribution. Lift variations were relatedto vortex shedding for different gap ratios. The average number of peaks in the cycleof lift force trace was used to estimate the Strouhal number of flow. Their resultsindicated that the vortex shedding persisted for smaller values of gap ratio as KCdecreased. They also reported that the Strouhal frequency increased by 50% for verysmall gap ratios of that for isolated cylinder. Sumer et al. [1991] obtained the gapratios at which vortex shedding was suppressed for different KC numbers estimatedfrom flow visualization and from lift force traces.

Dadman and Narayanan [1995] measured in-phase pressure distributions aroundthe horizontal yawed cylinder near a bed using a transducer. They measured thepressures at 18 points around the cylinder. The pressure patterns are found to repeatthemselves from one cycle to another except for some occasional blips.

Naeeni and Narayanan [2005] measured the wall pressure field on a cylinderoscillating over a plane bed in still water. Two gap ratios between the cylinder andthe bed (1.0 and 0.1) were considered in their study. Pressures on the peripheryof the cylinder were measured in 36 points by a pair of transducers. The pressurepatterns were found to be essentially repeatable from one cycle to the next. Theforces determined from the pressure profiles compared well with those measureddirectly by a force transducer.

In this paper, the spectral contents of the pressure fluctuations in different posi-tions on the cylinder have been studied in detail, with respect to the KC number and

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the bed proximity. Wavelet analysis is used for the simultaneous time–frequency rep-resentation of the pressure fluctuations in order to reveal the time at which abruptvariations in the spectral content and flow characteristics are experienced.

2. Experimental Details

2.1. Experimental apparatus

The experimental details have been fully described in the paper by Naeeni andNarayanan [2005] and only a brief description is presented here.

The experiments were carried out in the Department of Civil and StructuralEngineering, University of Manchester, United Kingdom [Naeeni, 2003]. The exper-imental apparatus consisted of an oscillating cylinder of 50 mm diameter kept atdifferent gaps above a plane bed in still water. The motion of the oscillatory cylin-der in the experiments was purely sinusoidal. u(t), the instantaneous velocity of thecylinder is expressed in Eq. (1)

u(t) = Um cos(ωt), ω =2πT. (1)

In which, Um is the maximum velocity, ω is the angular frequency and T is theperiod of oscillation.

A schematic diagram of the apparatus is shown in Fig. 1.Two pressure transducers were fixed from the inside to the wall of the cylinder

in a diametrically opposite orientation as shown in Fig. 2. The cylinder could be

1- Connecting rods of carriage 2- Guide rods 3- Cross beams4- Support beams 5- Swivel mount arrangement 6- Air bearing

7- Aluminum leg8- Tank wall9- Plate support beams10- Plastic flat plate11- Test cylinder

Fig. 1. Experimental apparatus.

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P01

P02

P03

P04

P05

P06P07

G

P08P09P10P11P12

P13

50 mm

D =

P15P14

P16

P17

P18

P19

P20

P21

P22

P23P24

P25P26

P27 P28 P29 P30P31

P32P33

P34

P35

P36θ

0°

90°

180°

270°

360°

Pressuretransducer A

Pressuretransducer B

Fig. 2. Positions of pressure measurement around the cylinder.

rotated to locate the transducers at 36 angular positions of 10◦ intervals (Fig. 2).Even though only two pressure transducers were used at a time, extreme care wastaken to match the phase of the oscillations and the phase of the pressure patterns.

2.2. Experimental conditions

Experimental conditions are presented in Table 1 in terms of the KC number, thegap ratio (G/D), the Reynolds number (Re), the ratio between Re and KC (β).Also included in the Table are the maximum velocity of the cylinder (Um), thetemperature at which the experiments were performed (Temp) and the period ofoscillation (T ). Each experimental run was carried out for 40 cycles of oscillation ofthe cylinder and the pressure around the cylinder were recorded 256 times in eachcycle.

Table 1. Summary of the test conditions for pressure measurements.

KC G/D Re β = Re/KC Um (m/s) Temp (◦C) T (s)

15 0.1,1.0 9.5 × 103 630 0.19

20 0.1,1.0 1.3 × 104 640 0.25

25 0.1,1.0 1.6 × 104 640 0.31 between 17 to 23 4

30 0.1,1.0 1.9 × 104 640 0.38

35 0.1,1.0 2.3 × 104 650 0.44

40 0.1,1.0 2.6 × 104 650 0.50

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3. Temporal Variation of Pressures

The dimensionless pressure at the angular position of θ is expressed as a pressurecoefficient Cp(t) defined in Eq. (2)

CP (t) =p(t) − p0

12ρU

2m

, (2)

where p(t) is the pressure at the point, p0 is the pressure at the same point in stillwater and ρ is the density of water.

Figure 3 shows typically the temporal variations of pressure coefficient p(t) atfour positions (Fig. 2) on the cylinder for KC number of 30 and gap ratio (G/D) of1.0. The time (t) plotted along the horizontal axis is normalized with respect to theperiod of oscillation (T ).

Figure 3 shows that the pressure patterns repeat themselves sensibly from onecycle to the next in this complex flow. This observation is the same as the conclusion

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3

-2

-1

0

1

2

3

4

Cp

000

0

G/D=1 Kc= 30

t/T

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3

-2

-1

0

1

2

3

4

Cp

090

0

G/D=1 Kc= 30

t/T

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3

-2

-1

0

1

2

3

4

Cp

180

0

G/D=1 Kc= 30

t/T

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3

-2

-1

0

1

2

3

4

Cp

270

0

G/D=1 Kc= 30

t/T

Fig. 3. Variation of pressure coefficient at four angular positions of the cylinder.

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of Isaacson [1974], Sumer et al. [1991] and Naeeni and Narayanan [2005] in theirinvestigation of pressure distribution on horizontal cylinder in oscillatory motion.

4. Spectral Analyses of the Pressure Coefficients

4.1. Fourier analysis

The normalized frequency spectra of pressures at various positions around the cylin-der have been investigated for different KC numbers and gap ratios using FastFourier Transform (FFT). Figure 4 displays typically the spectra at 36 positionsaround the cylinder for G/D = 1.0 and KC = 30.

Figure 4 shows that the fluctuations of pressure on the cylinder at θ = 0◦ and180◦ (see Fig. 2) are dominated by the main oscillation. It is predominantly theeffect of wake reversal and acceleration of the oscillatory cylinder. At θ = 90◦ and270◦ the dominant frequency of the pressure measurements is twice the frequencyof oscillation of the cylinder which is also noticeable in Fig. 3. The velocity field andthe vortex shedding contribute to the second harmonic of the pressures.

In Fig. 5 are shown the normalized frequency spectra of pressures around thecylinder for G/D = 0.1 and KC = 30. As can be seen, the first and second normalizedfrequencies are again dominant for different positions but in some positions on thecylinder, higher frequencies like the 4th harmonic are also present.

Although the general trend in terms of frequency distribution around the cylinderis almost the same for all KC numbers and gap ratios, the length along the perimeterof the pipe where each frequency is dominant (which is called the span of eachfrequency hereafter) is different for each case. To investigate such differences in eachcase more closely, the spans are related to the dominant frequencies around theisolated and near bed cylinders and are displayed in Figs. 6(a)–6(f) and Figs. 7(a)–7(f) for G/D = 1.0 and G/D = 0.1, respectively. In these figures, the frequency ismade dimensionless using the oscillatory frequency of the cylinder.

For the gap ratio (G/D) of 1, the flow characteristics are essentially the sameas those of an isolated cylinder. For G/D = 1.0, the first and second harmonics aredominant and the spans of each dominant frequency are almost the same for all KCnumbers studied (Fig. 6). However, a slight decrease in the span of second harmonicat the top and bottom of the cylinder could be observed for higher KC numbers.

The slight asymmetry in the distribution of the dominant frequency of fluctuatingpressure is observed that could be due to slight departure from one half cycle to thenext. Even though the oscillation of the cylinder is purely sinusoidal, the flow atthe end of each cycle can affect the flow in the following half-cycle. Some vorticesmight be swept back over the cylinder into the next half-cycle of flow. Furthermore,the experimental conditions in the tank housing, the oscillatory cylinder mechanismcan cause some asymmetry in the oscillating flow even though the tank used inour experiments is large enough to damp out disturbances farther away from thecylinder.

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0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/fW

P1θ=00

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/fW

P2θ=3500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/fW

P3θ=3400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/fW

P4θ=3300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/fW

P5θ=3200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/fW

P6θ=3100

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P7θ=3000

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P8θ=2900

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P9θ=2800

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P10θ=2700

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P11θ=2600

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P12θ=2500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P13θ=2400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P14θ=2300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P15θ=2200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P16θ=2100

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P17θ=2000

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P18θ=1900

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P19θ=1800

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P20θ=1700

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P21θ=1600

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P22θ=1500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P23θ=1400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P24θ=1300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P25θ=1200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P26θ=1100

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P27θ=1000

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P28θ=900

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P29θ=800

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P30θ=700

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P31θ=600

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P32θ=500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P33θ=400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P34θ=300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P35θ=200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f/f W

P36θ=100

Fig. 4. Normalized frequency spectra of fluctuating pressures at different positions around thecylinder for G/D = 1.0 and KC = 30, fw = frequency of oscillation.

For the near bed cylinder (G/D = 0.1) higher frequencies in pressure traces areobserved at the top and bottom of the cylinder (Fig. 7). The spans of the secondharmonic have considerably reduced at the bottom of the cylinder. The closeness ofthe bed has affected the vortex formation and shedding causing higher harmonics

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0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P1θ=00

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P2θ=3500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P3θ=3400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P4θ=3300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P5θ=3200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P6θ=3100

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P7θ=3000

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P8θ=2900

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P9θ=2800

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P10θ=2700

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P11θ=2600

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P12θ=2500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P13θ=2400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P14θ=2300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P15θ=2200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P16θ=2100

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P17θ=2000

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P18θ=1900

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P19θ=1800

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P20θ=1700

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P21θ=1600

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P22θ=1500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P23θ=1400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P24θ=1300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P25θ=1200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P26θ=1100

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P27θ=1000

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P28θ=900

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P29θ=800

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P30θ=700

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P31θ=600

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P32θ=500

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P33θ=400

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P34θ=300

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P35θ=200

0 1 2 3 4 5 6 7 80

0.20.40.60.8

1

f /f W

P36θ=100

Fig. 5. Normalized frequency spectra of fluctuating pressures at different positions around thecylinder for G/D = 0.1 and KC = 30, fw = frequency of oscillation.

in the pressure fluctuations. The regular vortex shedding is inhibited because thewall-side shear layer cannot develop as strongly as the opposing shear layer leadingto a weak interaction between the shear layers. The boundary layer separating asshear layer roll up into small vortices and provide higher harmonics. Forces exerted

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(a) (b)

(c) (d)

(e) (f)

Fig. 6. The dominant frequencies spans around the cylinder, G/D = 1.0, fw = frequency ofoscillation.

due to pressure on the cylinder are computed by integrating the temporal varia-tion of pressures around the periphery of the cylinder. The high frequency contentin the pressure traces especially along the top and bottom of the cylinder influ-ence the ultimate forces on the pipeline and are important from this view point.It is also desirable to know at what phase of oscillations these high frequency con-tents are formed. For this purpose, the continuous wavelet transform (CWT) isused.

4.2. Wavelet analysis

Fourier analysis is extremely useful for time series in which the frequency content isof great importance but in transforming to the frequency domain, time information

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(a) (b)

(c) (d)

(e) (f)

Fig. 7. The dominant frequencies spans around the cylinder, G/D = 0.1, fw = frequency ofoscillation.

is lost which is considered as a serious drawback of this method. Fourier transform(FT) gives the frequency information of the signal, which means that it tells us howmuch of each frequency exists in the signal, but it does not show the time whenthese frequency components exist. For stationary signals in which signal propertiesdo not change over time, this drawback is not important but for nonstationarysignals, transitory characteristics are often the most important part to show whenthese particular events took place. Hence, FT is not suited to detect them in timedomain.

Wavelet transform (WT) is capable of providing the time and frequency infor-mation simultaneously, hence giving a time–frequency representation of the signal.

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For the present study, wavelet analysis has been used to reveal the variation offrequency content in time for an average oscillation cycle.

4.2.1. Definition of the CWT

The CWT uses the inner product of the signal and a shifted and compressed orstretched versions of a wavelet function to measure the similarity between them.Stretching or compressing the wavelet corresponds to the physical notion of scale.The comparison of the signal and the wavelet at different scales and positions givesthe CWT coefficients. For a scale parameter (s) and position (u) along the timeaxis, the CWT coefficient is defined in Eq. (3) [Mallat, 2009]:

C(s, u) =∫ ∞

−∞f(t)

1√sψ∗

(t− u

s

)dt. (3)

In which f(t) is the signal, ψ is the wavelet function and * denotes the complexconjugate. The mother wavelet function is the basic form of the wavelet from whichdilated and translated versions are derived and used in the WT. Multiplying eachcoefficient by the corresponding scaled and shifted wavelet yields the constituentwavelets of the original signal. The CWT coefficient for each position and scale rep-resents how closely correlated the wavelet is with that section of the signal. Thelarger the coefficient C is in the absolute value, the more the similarity. The abrupttransitions in signals and the discontinuities could be detected using continuouswavelet analysis. CWT is also capable of detecting smoother signal features. Largewavelet coefficients are produced at scales where the oscillation in the wavelet cor-relates best with the signal feature, thus presenting the corresponding scale of theoscillation in time.

Over the past decade, the WT has emerged as a particularly powerful tool forthe elucidation of fluid signals [Addison, 2002]. Due to their high resolution in thewavelet domain, continuous wavelets are normally employed for feature detection inflows with recognizable coherent structures. The Mexican hat wavelet is often usedwhen compactness in the time domain is important. For the present investigation,the Mexican hat wavelet, which is very good at illustrating many of the propertiesof CWT analysis, has been chosen. The Mother Mexican hat wavelet function isdefined in Eq. (4) and shown in Fig. 8.

ψ(t) = (1 − t2)e−t2/2. (4)

In wavelet analysis, higher scales correspond to coarser signal features with lowerfrequencies and vice versa. While there is a general relationship between scale andfrequency, no precise formula exists. As it is desired to map between a wavelet ata given scale with a specified sampling period to a frequency in Hertz, a pseudo-frequency corresponding to each scale is defined in a general sense according to

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-4 -3 -2 -1 0 1 2 3 4-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Mexican hat wavelet function

Fig. 8. The mother Mexican hat wavelet function.

Eq. (5)

Fa =Fc

a · ∆ . (5)

In which “a” is a scale, “∆” is the sampling period, “Fc” is the center frequencymaximizing the FT of the wavelet modulus in Hz and “Fa” is the pseudo-frequencycorresponding to the scale a, in Hz.

4.2.2. CWT of pressure traces above and beneath the cylinder

Previous researchers have shown that the abrupt changes in the pressure distribu-tions are related to vortex shedding or to vortices being washed over the cylinder inthe reversal phase. The vortex shedding leads to abrupt changes in the time seriesand in the frequency content of the signal. These changes in the pressure distributionare associated with the appearance of a peak in the lift force trace [Sumer et al.,1991]. As WT is capable of representing the signal in simultaneous time/frequencydomain, we can detect the time of frequency changes attributed to the vortex shed-ding. This representation is a new information we get from WT which could not beachieved directly from the usual spectral analyses.

The average of pressure values measured in 40 cycles at the top and bottom ofthe cylinder are interrogated by CWT. FT analysis of fluctuating pressures alongthe cylinder has shown that the high frequency contents and the abrupt transitionsof pressure are observed mostly along the top and bottom of the cylinder. Hence,

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the pressure traces at these two positions are analyzed with WT. CWT coefficientsfor the fluctuations of pressures at the bottom and top of the cylinder are shown inFigs. 9 and 10, respectively, for G/D = 1.0 for different KC numbers. In these figures,the ordinates are wavelet scale and the corresponding pseudo-frequency shown onthe left and right of the graph, respectively. The abscissa is the time normalizedwith the oscillation period. The continuous wavelet coefficient is shown in black forextremum values of the CWT coefficient and white for zero values.

The oscillatory flow of the present investigations consists of four different phases.At the beginning of the flow cycle, the cylinder moves at maximum velocity. Thecylinder comes to rest after a quarter of the oscillation period. Then the velocitydirection reverses and the cylinder attains its maximum velocity at the next quarterof the oscillation cycle. The next half of the oscillation is like the first half but in thereverse direction. The scrutiny of the present results shows that when the cylinderreverses its motion, the pressure fluctuations are not in the same phase with thevelocity variations showing that the pressures do not respond instantaneously tothe velocity changes. The results show that the increase in KC number increasesthe time lag between velocity field and the pressure changes at the bottom of thecylinder (Fig. 9).

The asymmetry between the half cycles is seen in the results which are due to theasymmetry of vortex shedding between the half periods, as reported by Williamson[1985] and Sumer et al. [1991]. The asymmetry reduces with the increase in the KCnumber.

Sumer et al. [1991] observed that the stagnation point moves to the lee sidebefore the flow reverses. This behavior could be inferred from the spectral results aswell. In Fig. 9, it is seen that the average frequencies around the second harmonicof the oscillations all occur just before the flow reversal and also before the flowreaches its maximum velocity at the middle of the oscillations.

The WT analysis has been performed on the pressure traces at the top of thecylinder for G/D = 1.0. The results are shown in Fig. 10. At the top of the cylinder,the dominant frequency is again around the second harmonic of the main oscillation.When the cylinder reverses its motion the abrupt changes arising are detectable inthe results. The asymmetry between the half cycles also exists for this point atthe top of the cylinder and it decreases with the increase of the KC number. Thedominant harmonics occur again just ahead of the flow reversal.

The effect of bed proximity on the spectral behavior of the pressure fluctuationshas been investigated for the position at the bottom of the cylinder for the case ofG/D = 0.1. The results are shown in Fig. 11. As seen in the figure, the bed proximityexerts a significant influence on the pressure patterns at the bottom of the cylinderespecially at lower KC numbers. The contributions of higher frequencies are clearlyseen for lower KC numbers. For KC number of 15, the higher frequencies occur justbefore and after flow reversal which could very well be due to vortices detaching fromthe shear layer and being washed over the cylinder in the reversal phase. For KC

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Values of Wavelet Coefficients, G/D=1 KC=15

t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w

(a) (b)


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w

(c) (d)


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91P

seud

o fr

eque

ncy

f/f w

(e) (f)

Fig. 9. CWT coefficients for pressure fluctuations at the bottom of the cylinder, G/D = 1.0,extremum values in black and zero values in white, fw = frequency of oscillation, T =period of oscillation.

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t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f

/f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f

/f w

(a) (b)


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f

/f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f

/f w

(c) (d)


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f

/f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f

/f w

(e) (f)

Fig. 10. CWT coefficients for pressure fluctuations at the top of the cylinder, G/D = 1.0, extremumvalues in black and zero values in white, fw = frequency of oscillation, T = period of oscillation.

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Values of Wavelet Coefficients, G/D=0.1 KC=15

t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w

(a) (b)


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w

(c) (d)


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91

Pse

udo

freq

uenc

y f/

f w


t/T

scal

es a

0 0.25 0.5 0.75 1

10

20

30

40

50

60

70

6.4

3.2

2.13

1.6

1.28

1.07

0.91P

seud

o fr

eque

ncy

f/f w

(e) (f)

Fig. 11. CWT coefficients for pressure fluctuations at the bottom of the cylinder, G/D = 0.1,extremum values in black and zero values in white, fw = frequency of oscillation, T = period ofoscillation.

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number of 20, higher frequencies are observed at the time when the cylinder reachesits maximum velocity. The abrupt changes in the pressure at this time could be dueto shedding of the vortices. The sudden changes in the pressure distribution resultingfrom this phenomenon were previously reported by Sumer et al. [1991] supportedby flow visualization of the vortex shedding process. Their results indicated that theflow pattern and the pressure distribution change significantly because of the closeproximity of the boundary where the symmetry of formation of vortices breaks down.

The results show that at lower KC numbers, there is no specific pattern to thecontribution of the higher frequencies but as the KC number increases, the regularbehavior of the flow and pressure patterns become pronounced. The presence of thebed prevents the symmetry and regularity in the flow patterns especially at lowerKC numbers.

5. Conclusion

The pressure field, directly affects the forces exerted on the cylinder and henceis of great importance. In this paper, the pressure fluctuations around a cylinderoscillating in still water were analyzed for different KC numbers and gap ratios. Thedominant frequency of these fluctuations were investigated in 36 points around thecylinders and the spans of each dominant frequency are depicted around the cylinder.The dominant frequencies in the pressure fluctuations were detected using the FT.To reveal the time of occurrence of each frequency, WT was used by which, thefluctuations could be analyzed in time and frequency domains simultaneously. Thespecific conclusions obtained, are listed as follows:

• The fluctuations of pressure on the cylinder at positions 0◦ and 180◦ that coincidealong the line of motion of the cylinder are dominated by the main oscillation.The dominant frequency of the pressure fluctuation on the top and bottom of thecylinder is twice the frequency of oscillation of the cylinder. The velocity field andthe vortex shedding contribute to the second harmonic of the pressures.

• A slight asymmetry in the distribution of the dominant frequencies over the cylin-der is observed which is due to the asymmetry of each half cycle to the next.

• For the near bed cylinder (G/D = 0.1), higher frequencies in pressure traces areobserved in the top and bottom of the cylinder. The spans of the second harmonicare considerably reduced at the bottom of the cylinder. There is no clear cutvortex formation except that the separated boundary layer rolls up into a numberof small scale vortices.

• At the end of one cycle of the cylinder motion, there is a phase difference betweenthe pressure and velocity variations. The results show that the increase in KCnumber increases the time lag between velocity field and the resulting pressurefield on the cylinder.

• The asymmetry between the half cycles is seen in the continuous wavelet resultswhich are due to the asymmetry of vortex shedding between the half periods, as

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reported by Williamson [1985] and Sumer et al. [1991]. The asymmetry reduceswith the increase in KC number.

• The wavelet analysis shows that the average frequencies around the second har-monic of the oscillations occur just before the flow reversal, and also before theflow reaches its maximum velocity at the middle of the oscillations. This obser-vation could be deduced from the motion of the stagnation point to the lee sidebefore the flow reversal as reported by Sumer et al. [1991].

• The wavelet results for near bed cylinder show the influence of bed proximity onthe pressure at the bottom of the cylinder especially in lower KC numbers. Thecontributions of higher frequencies are seen for lower KC numbers.

• For a near bed cylinder oscillating with the KC number of 15, the higher frequen-cies occur just before and after the flow reversal which could be attributed to avortex detaching from the shear layer and rapidly being washed over the cylinderin the reversal phase. For KC number of 20, higher frequencies are observed atthe time when the cylinder is at its maximum velocity which could be due toshedding of the vortices at that time.

• The existence of a plane bed beneath a cylinder reduces the symmetry and reg-ularity in the flow patterns especially at lower KC numbers. As the KC numberincreases, the regular behavior of the flow and pressure patterns is progressivelyestablished.

References

Addison, P. S. [2002] The Illustrated Wavelet Transform Handbook, Introductory Theory and Appli-cations in Science, Engineering, Medicine and Finance (Institute of Physics Publishing, Bristoland Philadelphia).

Dadman, R. & Narayanan, R. [1995] “Pressure distribution on a circular cylinder oscillating close toa plane bed,” in Proc. 14th Int. Conf. Offshore Mech. and Arct. Eng., ASME, Vol. 1, pp. 411–423.

Isaacson, M. [1974] “The Forces on Circular Cylinders in Waves,” Dissertation, University ofCambridge.

Mallat, S. [2009] A Wavelet Tour of Signal Processing (Elsevier, United States).Matten, R. B., Hogben, N. & Ashley, R. M. [1979] “A circular cylinder oscillating in still water

in wave and in currents,” in Mechanics of Wave-Induced Forces on Cylinders, ed. Shaw, T. L.(Pitman, London), pp. 475–489.

Naeeni, S. T. O. [2003] “Force on Yawed Circular Cylinder Oscillating over a Plane Bed in Current,”Dissertation, UMIST, Manchester, UK.

Naeeni, S. T. O. & Narayanan, R. [2005] “Influence of plane bed on the force exerted on a cylinderoscillating in still water,” Ocean Eng. 3, 2217–2230.

Sarpkaya, T. [1976] “Forces on cylinders near a plane boundary in a sinusoidally oscillating fluid,”J. Fluids Eng., ASME 98(3), 499–504.

Sarpkaya, T. & Wilson, J. R. [1984] “Pressure distribution on smooth and rough cylinders in har-monic flow,” in Proc. Ocean Structural Dyn. Symp., September, 1984 (Oregon State University)pp. 341–355.

Sumer, B. M., Jensen, B. L. & Fredsoe, J. [1991] “Effect of plane boundary on oscillatory flowaround a circular cylinder,” J. Fluid. Mech. 225, 271–300.

Williamson, C. H. K. [1985] “Sinusoidal flow relative to circular cylinders,” J. Fluid. Mech. 155,141–174.

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spectral features of the pressure distribution around a cylinder oscillating in still water.pdf

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