Measurement of two-point correlations of velocity near a circular cylinder shedding a karman vortex street

MEASUREMENT OF TWO-POINT CORRELATIONS OF VELOCITY NEAR A

CIRCULAR CYLINDER SHEDDING A KARMAN VORTEX STREET

by M. Y. el BAROUDI

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JANUARY, 1960 UTIA TECHNICAL NOTE NO. 31

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CORRELATIONS OF VELOCITY NEAR A CIRCULAR CYLINDER SHEDDING A

J ANUARY, 1

wm

KARMAN VORTEX STREET

by

M. Y. el BAROUDI

UTIA TECHNICAL NOTE NO. 31 AFOSR TN-60-835

ACKNOWLEDGEMENT

The author wishes to thank Dr. G. N. Patterson for the opportunity to pursue this investigation at UT IA .

The initiation and supervision of the project by Professor B. Etkin and Dr. H. S. Ribner are gratefully acknowledged. Thanks are also due to Mr. L. N. Wilson, Mr. R. Keefe and Mr. G. Ludwig for their assistance.

Financial support for the work was provided by the

Defence Research Board of Canada under DRB Grant No. 9551-02 and by the United States Air Force under Contract No. AF 49(638)-249, the

latter monitored by A. F. Office of Scientific Research of the Air Research and Developrnent Commando

'

.

SUMMARY

Results of an experimental investigation of two-point correlations of velocity near a circular cylinder shedding a Karman vortex street are presented. The measurements were made along a line parallel to the generator of the cylinder which is at 900 _{fro.m the} upstream direction. The cylinder was mounted transversely in an air-stream, and two hot-wire probes were used as anemometers.

The curves of correlation coefficient versus probe separation approach zero as the separation between the probes is in-creased to large values.

A plot of correlation length versus Reynolds number. based in part on a conservative extrapolation. is also presented and compared with correlation length data from pressure measurements taken from reference 3.

(i) TABLE OF CONTENTS NOTATION 1. 2. 3.

4.

5. INTRODUCTION

BASIS FOR THE EXPERIMENTAL INVESTIGAT'ION 2.1 The Correlation Coefficient

2.2 The Correlation Length EXPERIMENTS

3.1 The Apparatus

3. 1. 1 General description of apparatus 3. 1. 2 The corre1ator of the hot wire set 3.2 Experimental Procedures REDUCTION OF RESULTS CONCLUSIONS REFERENCES FIGURES 1 TO 20 ii 1 2 2 2 2 2 2 3 4 )4 5 7

A B d F I I R.N. R. (x) 1) x 8 NOTATION

constant relating flow velocity to output signalof a

hot-wire channel for a given probe (.fPl:. / ma.. )

constant relating I to IT for a given probe

diameter of test cylinder

attenuation factor of thermal meter of the hot-wire set analyzer

instantaneous output signal from a hot-wire set channel

for a given probe.

,mean value of instantaneous output signal fr om a hot-wire

set channel for a given probe

outputs of channels 1 and 2 of the hot-wire set analyzer,

proportional to ~ and ~

ut

2. resp.

outputs of channels A and B of the hot-wire set analyzer,

proportional to the r. m. s. value of (UI' +- u I ")and (UI' - u,")

resp.

Reynold's number

measured correlation coefficient

instantaneous flow speed mean flow speed

perturbation velocities at the two probes due to the

flow-induced velocity field about the test-cylinder probe separation

angle measured from upstream direction

(1) 1. INTRODUCTION

For a range of Reynolds numbers a circular cylinder placed transversely in an air stream wil! shed a periodic array of vortices - the familiar Karman vortex street. The Aeolian tones produced by this shedding and related phenomena have been under investigation at U. T. 1. A. (Refs. 1, 2, 3).

At any instant the shedding vortex sheet wil! not extend the fuH length of the cylinder: it appears that segments of much smalle.r length tear off. It is the average coherent length of the shedding vortex sheets that figures in the calculated strength of the Aeolian tones. This average length can be obtained by a procedure involving two-point

correlations of pressure or velocity along the cylinder: the area under the curve of correlation versus separation is the effective shedding length.

To this end, measurements of two-point correlations of pressure along a cylinder were carried out in reference 3. However thecorrelations therein approached a finite asymptotic value at large separations rather than the zero value suggested by the physical

picture. In order to derive therefrom a finite correlation length it was necessary to guess at the correct asymptotic behavior of the curves.

It was suspected that the finite asymptotic pressure correlation might be due to a wind tunnel interference effect in the form of.a sound field. Thus a measurement of two-point correlations of the veloci ty field near a cylinder wouldbe relatively: inaensitive to such an interference. Such an experiment has been carried out with the particu-lar aim of checking the asymptotic correlation at particu-large probe separations, and of determining therefrom valid yalues of the average shedding along the cylinder (correlation length) of the vortices. The workwas carried out in an open jet, and the details are reported herein.

The results are presented in the form of curves of corre-lation coeificient versus probe separation for various Reynolds numbers, and as a plot of correlation length versus Reynolds number on which velocity and pressure (reference 3) correlation lengths are compared.

2. BASIS FOR THE EXPERIMENTAL INVESTIGATION 2. 1 The Correlation Coefficient

The. correlation coefficient, R(x) of two velocity signals is defined as:

(1)

and is a numerical measure of the coherence between the two velocity signals u,'

l'J,

t)

and u ," (

~,

t) ::.

U I'

l

~+

oe:..

I\;

)

2. 2 The Correlation Length

The correlation length,

~

• is a characteristic length

associated with the average size ,of the eddies being shed from the cylinder and it is given by:

=

(2)

3. EXPERIMENTS 3. 1 The Apparatus

3. 1. 1 General Description of Apparatus

All the experiments were conducted in the U. T. 1. A. duct facility, which is basically an open circuit' wind-tunnel. The flow is generated by a 7,800 c. f. m , blower at the inlet. T1;le blower is belt dr.iven by a variabie speed 10 h. p D. C. motor providing a continuous range from 0 to 2, 200 r. p . .m. An electronic counter is used to measure the blower r . p.m.

For this experiment an open jet was achieve'd by detaching the duct just downstream of the settling chamber. The dimensions of the jet are 12" by 8" at the .contraction cone exit (Figs. 1, 2).

The test cylinder was mounted in the jet close to the contraction cone exit by means of angle brackets fitted to the lips of the exit. The hot wire probes were attached to a second cylinder which in turn was .fittfd to the angle brackets aft (downstream of the test cylinder. The probes cou,ld be moved parallel and perpendicular to the flow, and in pitch.

(3 )

A Hubbard constant-temperatu:r:-e double channel hot-wire anemometer set was used in the correlation measurements in conjunction with two identical (conventionally built) hot-wire probes. The sensing elements of the probes were made of Tungsten wire (. 00025" diameter) (Figs. 3,4).

Two smooth-surface steel cylinders of 1/2" diameter and 1/4" diameter were used in theexperiments. An oscilloscope was used to observe continuously the velocity fluctuations from either probe.

3. 1. 2 The Correlator of the hot-wire Set

The Hubbard hot-wire set has non-linear circuits that can be adjusted to compensate the square-root relation between flow velocity and hot-wire heating current. Thus each hot-wire channel, with its probe connected, can be calibrated so that the output signal

cr)

is re-lated linearly to the mean flow velo city (U) over the operating velocity range:

U-=AI

(3 )

where A is a constant·(

+ps /ma )

for a given probe.

The velocity fluctuations are measured by a thermal meter in the analyzer section of the hot-wire set, whose output is related to the direct current in the wire by a constant factor B for a given probe. The thermal meter reading could be attenuated by factors F = 2,4, 6,8, 12,32.

(4)

where

=

the therm al meter reading in m. a.

I-I

=

the instantaneous deviation of the signal current from its mean value.

The perturbation velocity is related to the thermal meter reading through: (5a)

Combining equation (4) and equation (5a):

(5b) The perturbation velocities near the surface of the test cylinder (but outside the boundary layer) at a distance x apart along a generator (Fig. 5) are designated U,' and

U,"

.

The chosen generator lies along a side of cylinder 900 from the upstream centerline.

(6a) (6b) ] Z! I 11

-

!L-

+....!LJ....

(AI3F), (ABF)z. (6c)

[

J

-:z. \ 11 ~

-.1LL..

(6d) (A8F), (Ae,F)"

Squaring & subtracting equation (6d) from equation (6c) squared:

=

(ABF),

(ABF)1. [

IT~ - I~~

J

But: "1

1"1-..L

TT,.. -

TI} ~

r

T, IT~ 3.2 Experimental Procedures (7 ) (1) (8)

The experimental measurements of

IT

I

Ir

I

Iï

and

IT

~ were made ior each of the two rods tested at inc~easiÈg pro'be sepa-rations and decreasing flow speeds at each stepwise increase of probe separation.

The probe separation, x, was progressively increased from zero outwards from the centerline of the flow to as high as 13 dia-meters. All measurements were made at 9

=

900_{. The operating}

velocity range extended from 169 feet per second to zero feet per second.

The probes were calibrated at the beginning of every period of testing. Some of the tests were repeated on different days with no

significant variation in results. A pitot tube was used in the calibration.

Due to the narrowness of the jet cone, no valid results

could be .obtained ior large separations with the 1/2" cylinder. Due to the interference effects that arose when the two probes were in close proximity, no results were obtained at smal! separations with the 1/4" cylinder.

4. REDUCTION OF RESULTS

All the lT"

1T~

IITA and

Ir

measurements were reduced to give the correlation coefficient R(x) inl!>the form shown in equation (8).

(5)

A plot of the experimental correlation data versus Reynolds number was made (fig. 5) for various separations (R(x) versus Reynolds number for various

xl

d ). The circumstances mentioned at the end of section 3. 2 limited the measurements to

xl

d from 6 to 13 for the

Reynolds number range from 10, 000 to 22, 500 and to

xl

d from 1. 5 to 4.5 for the Reynolds number range from 22,500 to 45, 000.

The data for

xl

d = 1. 5, 2.5, 3.5, 4.5 was extrapolated over the range Reynolds number

=

22, 500 to 10, 000 on the basis of the definite pattern displayed by the experimental data for

xl

d

=

6, 9, 13

between Reynolds number - 22, 500 to 10, 000. (fig. 6).

Similarly the R (x) data for the

xl

d ::: 6, 9, 13 was extrapo-lated over the range of Reynolds number

=

22,000 to 45, 000 on the basis of the definite pattern displayed by the experimental data for

xl

d

=

1. 5, 2.5, 3.5, 4.5 between Reynolds number = 22,000 to 45, 000.

Curves of the correlation coefficient, R(x), versus

xl

d were plotted using the experimental data for

xl

d

=

1. 5, 2.5, 3.5, 4.5 between R. N.

=

22,000 and R. N.

=

45,000, for x/d

=

6, 7, 8, 9, 10, 11, 12, 13 between R. N.

=

10, 000 and R. N.

=

22, 500, and the extrapolated

data for

xl

d

=

6, 9, 13 between R. N.

=

22,000 and R. N.

=

45, 000, for

x/d=1.5, 2.5, 3.5, 4.5betweenR.N. =lO,OOOandR.N. -22,500

(figs. 7 to 19).

The curves obtained were then extrapolated to

xl

d

=

0 and

xl

d -70() and the areas under these curves were measured and the

correlation length, was obtained over the Reynolds number range used (see equation 2).

A curve of

~

lel

was plotted using velocity and pressure (reference 3) correlation data. (fig. 20).

5. CONCLUSIONS

The correlations of velocity at two points separated axia11y along the side of a cylinder shedding a Karman vortex street has been

measured. The correlation tends to zero at large separations.

This behaviour is in contradistinction to the behaviour of correlations of surface pressure in reference 3 which tended to a finite value at large separations. The present results support the inference of

a wind tunnel interference effect - presumably a sound field - in the

pressure measurements.

At a given Reynolds number the correlation obtained from

the pressure measurements (reference 3) are only of the order of sixty

It is not clear whether errors in the assumed correction in reference 3

of the asymptotic shape of the curves can account for the difference. It

is believed that the present correlation lengths based on velocity

mea-surements constitute the more reliable measurement of the effective

shedding lengths of the vort ex sheets.

The experimental correlations of pressure (reference 3) were obtained in a closed wind tunnel whereas the experimental

correlations of velocity (herein) were obtained in an open jet. It may be

worthwhile to make correlations of pressure and velocity under identical

physical conditions. Moreover, the relation between velocity and

pressure correlations might be examined by setting up a simple

theore-tical model of the flow (e.g. reference 2) in order to throw more light

1. 2. 3. Etkin, B. Korbacher, G.K. & Keefe, R. T. McGregor, D. M. Prendergast, V. ( 7) REFERENCES

Acoustic Radiation from a Stàtionary Cylinder in a Fluid Stream, U. T. 1. A. Report 39, 1956.

An Experimental Investigation of the

Oscillating Pressures on a Circular Cylinder in a fluid Stream, U. T . 1. A., Technical

Note, 14, 1957.

Measurement of Two-Point Correlations of the Surface Pressure on a Circular

I

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FIGURE 2

TEST CYLINDER

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- - U FIGURE 5 ~ ~ ~ 1 11

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FlGUR E 8 TWO-PQIN.T VELOCILY CORRELATION NEAR A CIRCULAR CYLINDER - RN=13. 520

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FROM VELOCITY CORRELATION MEASUREMENTS

0 _

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-<:J

- - - _ . _ - - - -10,000 20,000 30,000 40,000 RN

FIGURE 20 [PLOT OF CORRELATION LENGTH (NORMALIZED) VERSUS REYNOLDS NUMBER FOR A CIRCULAR CYLINDER

°

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