An Attempt to Measure Fluctuating Local Pressure in Free

8/11/2019 An Attempt to Measure Fluctuating Local Pressure in Free

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Abstract

We developed a technique for the measurement of fluctuating hydraulic-pressure at an arbitrary point in turbulent

flow. A pressure-measuring system was composed of a miniature static-pressure probe (SP-probe), a strain-gage-

type pressure transducer, and a tube connecting them. A model equation of dynamic response of the pressure-

measuring system was introduced, and a calibration technique for determining parameters in the model equation

was developed. Effects of the size of the SP-probe, length of the tube, and sensitivity of the pressure sensor were

systematically investigated using several SP-probes, tubes, and pressure transducers with different sizes, lengths,

and sensitivities, and fluctuating hydraulic-pressure measurement was performed in a wake of a circular cylinder.

Key words: Fluctuating pressure, Flow measurement, Wake, Pitot tube, Free shear flow

1. Introduction

Fluctuating pressure is a fundamental and important property of turbulent flow as well as fluctuating velocity, and ef-forts to measure the fluctuating static-pressure in turbulent flow have been made from the 1950s. However, the fluctuating

pressure measurement at an arbitrary point remote from wall is still difficult as compared to velocity measurement; one

needs to use a Pitot-tube-type pressure probe to pick up the instantaneous static-pressure with sufficient spatial resolution,

but the intrusive nature results in some problems such as disturbance to the flow and measurement error caused by lateral

velocity component. In addition, compressibility of air inside the pressure probe results in a non-flat frequency response.

These problems in experiment in air were addressed by Shirahama and Toyoda (1993) and Toyoda et al. (1994),

who optimized geometric configuration of a miniature static-pressure probe (SP-probe) so that the disturbance to the flow

and the cross-flow error were minimized. They also proposed some techniques to calibrate the frequency response of the

pressure-measuring system. The similar techniques have been adopted in several experimental studies (Iida et al., 1999;

Naka et al., 2006; Tsuji et al., 2007; Kawata et al., 2011; Terashima et al., 2012) and successful measurement results have

been reported.

On the other hand, the fluctuating static-pressure measurement in water is still limited to wall-pressure measurement,

while some aforementioned successful measurements have been reported by experiments in the air. The reason might

be difficulties in manufacturing a pressure sensor applicable to hydraulic pressure measurement and, at the same time,

sensitive enough to measure small pressure fluctuation produced by turbulence.

An attempt to measure fluctuating local pressure in free

turbulent flow in water

*Department of Mechanical Engineering, Keio University

3141 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 2238522, Japan

E-mail: [email protected]

Received 6 January 2014

Takuya KAWATA, Hoshito MAEDA and Shinnosuke OBI

In the past hydraulic wall-pressure measurement, a strain-gage-type pressure transducer was used as a pressure sensor

and connected to the pressure tap on the wall by tubing. Dynamic response of such a pressure-measuring system has been

investigated in the previous studies (e.g., Geddes et al., 1984), and it was pointed out that the dynamic response was non

flat mainly due to elasticity of the pressure sensor while effect of compressibility of the fluid is negligible. The methods

to correct these effects have been studied in literatures (Donovan et al., 1991; Taylor and Donovan, 1992; Donovan et al.,

1994; Aydin, 1998). In these studies, the behavior of the pressure-measuring system was modeled as that of a harmonic

oscillator with one degree of freedom, and parameters in the model equation, the natural frequency and the damping

factor, were determined by calibration.


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Fig. 1 (a) Schematics of pressure-measuring system; (b) its model.

In the present study, a technique for fluctuating pressure measurement at an arbitrary point in turbulent flow in

water was developed. A pressure-measuring system was composed of an SP-probe, a strain-gage-type pressure transducer

and a tube connecting them. A model equation of the behavior of the pressure-measuring system was introduced and

the parameters in the model equation were evaluated in a newly proposed calibration procedure. Dependency of these

parameters on size of the SP-probe, length of the tube and sensitivity of the pressure transducer was systematically

investigated. The fluctuating pressure measurement was performed in a wake of a circular cylinder, and correction of

pressure signal based on the calibration was demonstrated.

2. Analytical Model for Frequency Response of Pressure-Measuring System

The system, which enables us the fluctuating static-pressure measurement at an arbitrary point in water, is schemati-

cally shown in Fig. 1a. It comprises an SP-probe, tubing and a pressure transducer, and can be simplified as a tube which

connects the pressure sensor to a pressure source with sudden contraction at one end as shown in Fig. 1b. In the following,

an equation which relates the true value of fluctuating pressure at the pressure source ps and the measured value pm is

derived.

Flow inside the tube is driven by pressure difference between the pressure source and inside the cavity of the pressure

transducer, and it is damped by the viscosity of the fluid. Hence, the behavior of volume flow rate through the cross section

of the tubingQ can be modeled as:

IdQ

dt +RQ = (pm ps) , (1)

where Iis the fluid inertance, and R is the resistance of the system mainly caused by the viscosity of the working fluid.

The volume flow rate Q can be related to the volume change in the cavity of the pressure transducer Vas;

dV

dt= Q. (2)

Introducing the elastic compliance of the pressure transducer C, which represents the elasticity of a diaphragm in the

pressure transducer;

C= dV

dpm, (3)

and substituting Eqs. (2) and (3) into Eq. (1), one can obtain the equation which relates pm to ps as:

ICd2pm

dt2 +RC

dpm

dt+ pm = ps. (4)


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Fig. 2 Schematics of static-pressure probes.

Equation (4) has the same form as the conventional model equation of a system for the wall-pressure measurement (Dono-

van et al., 1991). Thus, the behavior of the pressure-measuring system is equivalent to that of a damped harmonic oscil-

lator, and the damping ratio and the natural frequency (in rad/s) of the system are given as

=R

2

C

I, n =

1

CI, (5)

respectively. Solving Eq. (4) with the pressure input fluctuating at a constant frequency, one obtains the amplitude ratio

Aand the phase lag between pmand ps as:

A= 1

(1 u2)2 + 42u2(6)

= tan1

2u

1 u2

(7)

withu being the non-dimensional frequency defined as u = /n. The profiles ofA and against the frequency of the

input pressure fluctuation were measured by calibration of dynamic response of the pressure-measuring system, and the

damping ratio and the natural frequency were evaluated by fitting measured variation ofA and to Eqs. (6) and (7).

3. Experimental Apparatus

3.1. Pressure-Measuring System

3.1.1. Static-Pressure Probe The SP-probes employed in the present study is schematically shown in Fig. 2. There

were four types of the SP-probe: the probe-air, d10, d15 and d20. The probe-air had similar geometric configuration to

those employed in previous experiments in the air (Toyoda et al., 1994; Naka et al., 2006). It consisted of a thin tube with

a circular corn tip, and the outer and inner diameters were 1.0 and 0.8 mm, respectively. There were four pressure-sensing

holes on the surface separated by 90 in the circumferential direction with the diameter of 0.4 mm.

The probe d10, d15 and d20 were designed for hydraulic pressure measurement, and had the inner diameters of the

tube part d of 1.0 mm, 1.5 mm, and 2.0 mm. The diameters and lengths of each probe are presented in Fig. 2. They

were made of transparent material so that one could check if bubbles remained inside of the SP-probe, and the tip was

semi-spherical shape. The number of the pressure-sensing holes was eight, and we determined their diameter , so that the

impedance of the pressure-measuring system was retained low. To accomplish this, we assumed the pressure loss caused

by the eight pressure-sensing holes to be equal to that in the tube of the SP-probe for the same volume flow rate. Under

the assumption that both the flow through the tube of the SP-probe and that through the eight pressure-sensing holes are


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20 10 0 10 200.04

0.03

0.02

0.01

0

angle of attack ()

Cp

=(p

p0

)/U2

Fig. 3 Error in pressure measurement due to cross flow: blue, probe-d20; green, probe-d15; red, probe-d10;

black, probe-air.

approximated by the Hagen-Poiseuille flow, the volume flow rate in the former and that through the latter are proportional

tod4 and 84, respectively:

84

d4 =1

d=

1

8

14

=0.59. (8)

Therefore, the diameter of the pressure-sensing holes was determined to be 60% of the tube diameter of the SP-probe.

The analysis based on the Hagen-Poiseuille flow would over-estimate the pressure loss caused by the pressure-sensing

holes, as there would be only minor viscous effect. Hence the diameter of the pressure-sensing holes determined above

should be large enough not to cause a major influence to the dynamic response of the pressure-measuring system.

Measurement error caused by the cross flow was experimentally explored. The SP-probe was placed in a free stream

of a uniform flow of a wind tunnel, and the pressure was measured with various angles of attack. Figure 3 shows the

variation of the pressure coefficient defined as

Cp = p p0

U2, (9)

with p0 being the pressure value measured with the flow angle of 0. It is shown that pressure is underestimated when the

flow direction is not parallel to the axis of the SP-probe. The probe-air shows better angle characteristics than those of the

SP-probes designed for use in water, which may be attributable to difference in shape of the tip section.

3.1.2. Pressure Transducer A low-range strain-gage-type pressure transducer was used for pressure sensor in order

to sensitively measure the small pressure fluctuation in a low frequency range. In order to investigate the effect of the

elastic complianceC, three pressure transducers with differentCwere used for comparison; two DP45s and one DP103

(Validyne) were employed, and their characteristics are summarized in Table 1.

3.1.3. Tubing A plastic tube was used for tubing between the SP-probe and the pressure transducer placed inside and

outside the flow, respectively. The inner diameter and the thickness were 6 mm and 0.5 mm, respectively, and the length

was 0.96 m.

3.2. Other instruments and Data Acquisition

The experiments were undertaken in a closed-loop water tunnel with maximum free stream velocity of 0.28 m/s.

For the other hardware, a carrier modulator (PA501, Validyne) was used for amplification of the signals from the pres-

sure transducer, and an analog low-pass filter (DT-6LF2, NF CORPOLATION) and a 16 bits A/D converter (PCI-6621,

National Instruments) were used for filtering and acquisition of samples.

Table 1 Characteristics of pressure transducer; s, sensitivity;Vvolume of pressure cavity; Vmax, maximum

volumetric displacement.

Range (Pa) s(Pa/V) V (cc) Vmax (cc) C(mm3/Pa)

DP45-1 84 16.7 0.16 0.016 0.192

DP45-2 167 33.2 0.16 0.016 0.0967

DP103 28 5.7 0.57 0.057 1.96


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function

generator oscillator

piston pressure chamber

water channel

pressure transducer

pressure transducer (reference)

Fig. 4 Schematics of system for calibration of frequency response of pressure-measuring system.

4. Frequency response of Pressure-Measuring System

4.1. Response to Sinusoidal Pressure Fluctuation

Response of the pressure-measuring system against pressure input oscillating at a constant frequency was experi-

mentally explored. The system for this calibration is schematically shown in Fig. 4. The pressure-measuring system to

be calibrated was set up as it was in the main measurement; the SP-probe was placed in the water channel and connected

to the pressure transducer located outside by a tube which is identical to that used in the main experiment. The reference

port of the pressure transducer was connected to a chamber, and the oscillatory pressure input was generated inside the

chamber by driving a piston using an oscillator. The frequency and the amplitude of the pressure input were controlled by

a function generator. The true amplitude and the phase of the pressure input were monitored by another pressure trans-

ducer for reference, and the amplitude ratio A and the phase delay were evaluated by comparing the pressure signals

measured simultaneously by these two pressure transducers.The dumping ratioand the natural frequency n were evaluated based on the measured profiles ofA and . The

n was obtained as the frequency at which is /2 by interpolating the measured variation of. The dumping ratio

was evaluated as

=

1

1 A2max

2 (10)

where Amax is the maximum value ofA. Equation (10) cannot be used in the case where the measuring system is over-

damped and the profile ofA does not have a maximum value. In such case, the optimal value ofwas obtained by fitting

Eq. (6) to the measured profile ofA using fminsearch, which is function for optimization available in MATLAB R.

The frequency response of the pressure-measuring system was investigated for various combinations of the SP-probe

and the pressure transducer, and the values of and n are summarized in Table 2. The measured variation ofA andare compared for cases with DP45-1 and various SP-probes in Fig. 5 and for cases with the probe d20 and various

pressure transducers in Fig. 6. The circles stand for the measured values, and the solid lines represent the profiles obtained

by Eqs. (6) and (7) with the calibrated values ofand n in Table 2. Although only the measured values ofA and at

vicinity of the natural frequency were used for evaluation ofand n, the profiles drawn by Eqs. (6) and (7) are in good

Table 2 Dumping ratio and natural frequency of various pressure-measuring systems.R,IandCare given in

mm3/Pa, 102kg/mm4 and mm3/Pa.

DP45-2 DP45-1 DP103

SP-probe air d10 d15 d20 air d10 d15 d20 air d10 d15 d20

1.5 0.41 0.20 0.16 1.9 0.52 0.24 0.20 4.0 1.1 0.55 0.46

n 19.7 17.4 19.3 20.3 12.1 11.7 13.2 13.7 2.88 3.52 4.58 4.78

R 1.6 0.49 0.21 0.16 1.7 0.46 0.19 0.15 1.4 0.325 0.122 0.098I 2.7 3.4 2.8 2.5 3.6 3.8 3.0 2.8 6.1 3.8 2.4 2.2

C 0.0967 0.192 1.96


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101

100

101

0

1

2

3

A

101

100

101

0

0.5

1

Frequency (Hz)

(

rad)

Fig. 5 Frequency response of pressure measuring systems with DP45-1 and various SP-probes: blue, probe-d20;

green, probe-d15; red, probe-d10; black, probe-air. Circle plots and solid line show the measured values

and the fitted profiles, respectively.

agreement with all the measured values throughout entire the calibrated frequency range, which indicates that the model

Eq. (4) represents well the behavior of the present pressure-measuring system.

Comparing the frequency response of the pressure-measuring systems with different SP-probes shown in Fig. 5, one

can obviously see that the resonance peak magnitude ofA becomes smaller as the SP-probe becomes thinner. Especially

in the case with the probe-air, the system is overdamped; the profile ofA does not show any peaks and becomes large

already in the frequency range lower than 1 Hz. This significant damping is also indicated by the values ofin Table 2;

the value ofin the case with the probe-air is much larger than those of the systems with the other SP-probes. On the

other hand, the variations ofcrosses= /2 around 2 Hz regardless of the size of the SP-probe, indicating that the

natural frequencyn is not affected by the size of the SP-probe.

The results presented in Fig. 6 shows that the compliance Csignificantly changes both and n: asCincreases, the

natural frequency is shifted toward the lower frequency region, and also the system becomes more damped. This can beexplained by Eq. (5) asCappears in bothand n.

The fluid resistance R and the inertance Iwere also evaluated based on Eq. (5), and the values are summarized in

Table 2. One can see thatR significantly increases as the SP-probe becomes thinner. Hence, the size of the SP-probe

mainly affects the damping ratio through the value of resistance R, and does not the natural frequency n because R

does not appear inn (see Eq. (5)). On the other hand, Iis relatively insensitive to both the size of the SP-probe and the

compliance of the pressure transducer.

101

100

101

0

1

2

3

4

A

101

100

101

0

0.5

1

(

rad)

Frequency (Hz)

Fig. 6 Frequency response of pressure measuring systems with probe-d20 and various pressure transducers: blue,

DP45-2; green, DP45-2; red, DP103


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pressure transducer

pressure box

air

water

piston

soap bubble

Pa+ P

Pa+gh+Ph

Fig. 7 Schematics of system for step-response test.

4.2. Step-Response Test

Response to a step input was examined for further investigation on the dynamic response of the pressure-measuring

systems. A sudden pressure drop was generated using a system schematically sown in Fig. 7. The SP-probe of the

pressure-measuring system to be calibrated was fixed in the water phase in a pressure box. A T-shape tube was connected

to the air phase of the pressure box at one end and to a piston at another end. The other end was closed by a membrane

of soap solution. The pressure in the air phase in the pressure box was kept higher than the atmospheric pressure by the

piston, and sudden pressure drop was generated by breaking the membrane of soap solution.

The behavior of the measuring system against the step input is described by the following differential equation:

d2pm

dt2 + 2n

dpm

dt+ 2npm = 0. (11)

Equation (11) can be easily solved, and the analytical step response of the pressure-measuring system is obtained as:

pm

p0=

ent

cos qt+

n

qsin qt

( 1)

(12)

where p0 is the height of the step input, andq, p1 and p2 are respectively defined as:

q =n

1 2,

p1 =n(+

2 1),

p2 =n(

2 1).

Figure 8 presents the step response of the measuring systems with DP45-1 and various SP-probes, comparing those

measured and predicted by Eq. (12). The measured responses are indicated by the blue, green, cyan and red lines for the

probe-d20, d15, d10 and air, and it is shown that step response of the system with the probes-d10, d15 and d20 shows the

overshoot, but that of the system with the probe-air is overdamped and does not oscillate. The step responses predicted

by Eq. (12) with the values ofandn in Table 2 are indicated by the black dotted, dashed, chained and solid lines for

the probe-d20, d15, d10 and air. The predicted step response for the case with the probes-air and d10 are in quite good

agreement with the measured step responses, and those for the case with probes-d15 and d20 also agree fairly well withthe measured step response while certain underestimate of overshoot can be seen. This results also supports consistency

of the present calibration results.


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Table 3 Parameters of pressure-measuring system with different tubing length.

L(m) /0 n/n0 R/R0 I/I00.46 1.1 1.26 0.94 0.86

0.96 1 1 1 1

1.9 0.94 0.76 1.23 3.73

0 0.5 1 1.51

0.5

0

0.5

1

t(s)

p/p

0

Fig. 8 Comparison of measured and predicted step response of pressure-measuring system with DP45-1. Probe-

d20, blue and ; probe-d15, green and ; probe-d10, cyan and ; probe-air, red and .

4.3. Effect of Tube on Dynamic Characteristics of Systems

The present pressure-measuring system has tubing with certain length between the SP-probe and the pressure trans-

ducer unlike the systems used in the measurement in air, in which the transducer is directly embedded to the SP-probe.

The dynamic characteristics of such pressure-measuring systems may depend not only on length but also shape of the

tube. In this section, such effect of the tubing is addressed by the step-response test.

4.3.1. Effect of tubing length The effect of the tubing length was examined by comparing the step response for

different tubing lengths. Three tubes were used for comparison: 0.46 m, 0.96 m and 1.9 m. It should be mentioned that

the tube with length of 0.96 m was identical to that used in the main experiment.Figure 9 compares the step response of the pressure-measuring systems with the DP45-1, the probe-d20 and the three

different tubes. It is shown that the step response is clearly affected by the length of the tubing. The parameters of the

dynamic response for each case were evaluated by fitting Eq. (12) to the measured step response, and the obtained values

are summarized in Table 3. The values are scaled by those obtained in the case of the tubing length being 0.96 m. It is

shown that bothR andIincrease as the tubing length increases. The increase ofR is attributable to increase of the area of

inner wall of the tube, where the viscous drag acts on the oscillating fluid inside the tubing. The increase ofI is simply

interpreted as the increase of the fluid mass inside the pressure-measuring system.

The natural frequency of the systemn moves toward the lower frequency range as the tubing length increase due

to the increasing I(see Eq. (5)). The damping ratiois the function of both R and I. It decreases as the tubing length

increases because the variation ofIis more significant than that ofR. Thus, the tubing length affects both the damping

ratio and the natural frequency of the pressure-measuring system through the fluid inertance I.

0 0.5 1 1.5 2

1

0.5

0

0.5

1

t(s)

p/p0

L=0.96 m

L=0.46 m

L=1.9 m

Fig. 9 Step response of pressure measuring systems with probe-d20, DP45-1 and tubing with different lengths.


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0 0.5 1 1.51

0.5

0

0.5

1

t(s)

p/p

0

straight

loop

arch

Fig. 10 Step response of pressure measuring systems with same SP-probe, tubing, pressure transducer but

different shape of tubing.

4.3.2. Effect of tubing shape The effect of curvature of the tubing is examined. The step responses of the systems

comprising the same SP-probe, the same pressure transducer and the same tube but with different shape of the tube,

straight, looped and arched, were measured. Figure 10 shows that the step responses were identical to each other regardless

of the tubing shape, indicating that the tubing shape does not affect the dynamic characteristics of the measuring system.

5. Fluctuating Pressure Measurement in a Wake of a Circular Cylinder

5.1. Experimental Condition

Fluctuating pressure measurement was performed in a wake of a circular cylinder. The measurement was undertaken

in a closed-loop water channel. A circular cylinder with the diameter D of 20 mm was placed in a free steam of the water

channel, and the free stream velocity was fixed at U = 0.28 m/s. The temperature of the water was 39C during the

measurement. The origin of the Cartesian coordinates was placed at the center of the cylinder and the x andy axes were

taken in the streamwise and transverse directions, respectively. The Reynolds number based on Uand D,Re = UD/,

was 7800, and the measurements were conducted at x/D = 6.

The main SP-probe was placed in the wake, and the secondary SP-probe with geometric configuration identical to

the main probe was placed at the same streamwise location but outside the wake, in order to monitor the background noise

on the fluctuating pressure. The background noise on the pressure signal measured by the main probe was reduced based

on the signals measured by the secondary probe in the same manner proposed by Naguib et al. (1996). For the pressure

transducer, the DP45-1 was used with the SP-probes d15 and d20, but the DP45-2 was used for with the SP-probe d10

and air because of the better dynamic response. The sampling rate and sampling time at each measurement point were

100 Hz and 300 s, respectively.

The signals of fluctuating pressure were corrected based on the calibration results of the dynamic response of the

pressure-measuring system. Applying the Fourier transform to the measured pressure signals, we evaluated the complex

Fourier coefficientBn. The amplitude change and the phase delay atnth frequency fn, namelyAnand n, were evaluated

Fig. 11 Measurement condition.


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3 2 1 0 1 2 3

0.1

0.05

0

y/D

P/U2

Fig. 12 Distribution of mean pressure and atx/D = 6 in a wake of a circular cylinder: blue, d20; green, d15; red,d10; black, air.

by Eqs.( 6) and (7) with the values ofn and in Table 2. Using these values ofAn and n,Bn was corrected as;

Bn = Bn

An {cos(n) + j sin(n)}, (13)

where jis the imaginary unity, and the pressure signals were reconstructed by applying inverse Fourier transform as

p =Nn=1

Bnexp (j2fnt) . (14)

As the sampling time was 300 s, the frequency resolution of the Fourier transform was 1/300 Hz. Besides, a low-pass

filtering with the cut-offfrequency of 8 Hz was applied to the corrected pressure signals, in order to filter out the high-frequency noise. The FFT, the interpolation, and the low-pass filtering included in the above procedure were conducted

by using the functions fft, interp1, and filtfilt available in the MATLAB R library.

It should be mentioned that the measurement locationx/D =6 belonged to the flow region associated with relatively

strong velocity fluctuation which might contaminate the pressure measurement by the cross-flow effect as depicted in

Fig. 3. It is, however, difficult to validate the present pressure measurement results because of the lack in available data of

the pressure fluctuation. Instead, in the present study, a focus is put on the consistency among the proposed four different

pressure-measuring systems and the noise level against the signal. The accuracy of the pressure measurement under a

relatively strong velocity fluctuation was addressed in a separate experiment conducted in the air by Kawata et al. (2014).

5.2. Results and Discussions

The distributions of the mean pressure P at x/D = 6 measured by the four different SP-probes are presented inFig. 12. The values are scaled byU2, and the mean pressure values at y/D =3.0 are taken to be zero. It should be also

mentioned that the pressure in the range ofy/D< 2.0 was not measured by the probes d10 and air due to the limitation of

the traverse equipment. The profile ofP measured by all the SP-probes show symmetric profiles with respect to y/D = 0

and take the minimum of about 0.1U2at the wake center. The agreement between these results is reasonably good.

Figure 13a presents the distributions of the root-mean-square of the pressure fluctuation evaluated without correction

based on the calibration results. The values of the pressure fluctuation p are also scaled by U2. The profiles of the

uncorrectedp measured by the various SP-probes show much different distributions. Those measured by the probes d10,

d15, and d20 indicate the two peaks of the pressure fluctuation located beside the wake center, but the peak magnitude is

different from each other. On the other hand, the profile ofp measured by the probe-air shows much smaller value and

has no peak of fluctuation.

Figure 13b presents the profiles of the pressure fluctuation evaluated with the correction. The profiles of p

measuredby the four SP-probes are in good agreement with each other while the profiles evaluated without the correction were much

different from each other. In particular, the result measured by the probe-air was drastically changed by the correction;

two peaks of the fluctuation appears at y/D = 1.


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3 2 1 0 1 2 30

0.02

0.04

0.06

0.08

y/D3 2 1 0 1 2 30

0.02

0.04

0.06

0.08

y/D

pU28

pU28

(a) (b)

Fig. 13 Distribution of pressure fluctuation at x/D= 6 in a wake of a circular cylinder: (a) uncorrected,

(b) corrected. Colors indicate same SP-probe as in Fig. 12

Fig. 14 Power spectra density of pressure fluctuation measured aty/D= 1.0: (a) uncorrected, (b) corrected;

blue, d20; green, d15; red, d10; black, air.

Figure 14 presents the power spectrum density (PSD) of the fluctuating pressure at y/D = 1.0, comparing the

results measured by the various SP-probes. The profiles of the PSD of the uncorrected pressure are show in Fig. 14a. All

of the profiles measured by the four different SP-probes have a significant peak at f = 2.6 Hz, which corresponds to the

Strouhal numberSt= f D/U of 0.186. This peak was obviously caused by the Karman vortices shed from the cylinder.

The difference between the results is significant in the frequency range higher than 1 Hz, including the vortex shedding

frequency. The PSDs measured by the probe-d10, d15, and d20 have the broad hill, which was caused by the resonance.

On the other hand, the profile of the PSD measured by the probe-air does not show such broad increase because the system

with the probe-air is overdamped.

In Fig. 14b, it is shown that the PSDs of the measured pressure are in good agreement with each other after the

correction. The broad hill apparent in the profiles of the uncorrected PSD measured by the SP-probe d10, d15, and d20

disappear although the significant peak by the Karman vortex shedding still remains. The profile of the PSD measured

by the probe air also drastically changed by the correction. This agreement between the pressure fluctuations by the

correction supports validity of the present measurement technique.

In order to further examine the measurement results, the probability density function (PDF) of fluctuating pressure

was investigated at y/D = 1.0 and 3.0; the former was the location where the pressure fluctuation reached a local max-

imum, while the latter was outside the wake region where the pressure fluctuation is so weak that basically only the

background noise was measured. Figure 15 presents the PDFs at these locations measured by the four SP-probes after

the correction of the dynamic response was made. The results measured by the four different SP-probes agree well with

each other. The PDFs measured at the edge of the wake y/D = 3.0 are concentrated around p/U2 = 0, while the PDFs

measured at the peak location y/D = 1.0 show smooth distributions with a notable negative skewness. Both distributions

are smooth and free from scratch noise of any kind. Based on these observations, we consider that the noise level of the

pressure measurement was sufficiently small.


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0.2 0.1 0 0.1 0.20

10

20

30

40

50

p/U2

PDF

0.2 0.1 0 0.1 0.20

2

4

6

8

p/U2

Fig. 15 Probability density function of fluctuating pressure: (a) measured aty/D= 3.0, (b) measured at

y/D= 1.0. Colors indicate same SP-probe as in Fig. 12.

3 2 1 0 1 2 33

2

1

0

1

S

y/D3 2 1 0 1 2 3

3

2

1

0

1

y/D

S

(a) (b)

Fig. 16 Distribution of skewness factor atx/D= 6 in a wake of a circular cylinder: (a) uncorrected, (b) corrected.

The colors indicate same SP-probe as in Fig. 12

The appropriateness of the correction based on the present calibration results is also supported by the higher moment

of the pressure fluctuation. Figure 16 presents the distribution of the skewness factor of the pressure fluctuation, defined

as S p3/p3, comparing the profiles evaluated from the measured pressure signals before and after the correction.

The profiles ofSevaluated without the correction show the negative values of skewness across the wake, and have two

negative peaks located at y/D = 2. Among the results, those measured by the probe-air shows much larger magnitude

of the skewness than those measured by the other probes in the center region of the wake. On the other hand, the profilesofSevaluated with the correction are in fairly good agreement with each other, which also supports the appropriateness

of the correction on the measured pressure based on the calibration results.

As shown in the above discussion, the measured fluctuating pressure were examined from different aspects including

the second- and third-moments, PSD, and PDF, and these results measured by the four different pressure-measuring

systems showed a clear consistency with each other. These observations supports a fair reliability of the present method

of dynamic-response calibration.

6. Concluding Remarks

A technique for fluctuating static-pressure measurement at an arbitrary point in a turbulent flow in water was de-

veloped using a pressure-measuring system comprising a miniature static-pressure probe, a strain-gage-type pressuretransducer, and a tube connecting them. A model equation of behavior of the pressure-measuring system was introduced,

and a calibration technique for determining the parameters in the model equation was developed. Effects of the size of

the pressure probe, length of the tube, and sensitivity of the pressure transducer were systematically investigated in the


13/13

dynamic-response calibration. The fluctuating pressure measurement was performed in a wake of a circular cylinder, and

the pressure-related statistics measured by different four pressure-measuring systems, including the second- and third-

moments, power spectrum density, and probability density function, were in good agreement with each other after the

correction based on the dynamic-response calibration.

Acknowledgment

The authors are very grateful to Prof. Kuniaki Toyoda of Hokkaido Institute of Technology for his shrewd advices on

fluctuating pressure measurement. We are also grateful to Assoc. Prof. Koji Fukagata of Keio University for invaluable

discussions. This research was financially supported through Grant-in-Aid for Research Fellow (No. 23-3769) of Japan

Society for Promotion of Science.

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