Measurement of the Reynolds Stresses in a Circular Pipe as a Means of Testing a DISA Constant-Temperature Hot-Wire Anemometer

~iIII4.1,

iett,r••• !

T,I,

(~L'~)

1~

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~~

MEASUREMENT OF THE REYNOLDS STRESSES IN A'CrRCULAR PIPE AS A MEANS OF TESTING A DlSA CONSTANT-TEMPERATURE

HOT WIRE-ANEMOMETER by

R.P. Pate1 Tech. Note 63-6

Mechanica1 Engineering Research Laboratories MeGi11 University

Supported under D.R.B. Grant Number 9551-12

ACKNOWLEDGEMENTS

The author wishes to extend his sincere thanks to

Dr. B.G. Newman who advised him throughout the course of this work.

Thanks are also due to Mr. P.S. Moller who made the

experimental rig available for the present investigation.

The author is currently supported financially by a

Canadian Commonwealth Scholarship.

Many thanks are due to the Royal College, Nairobi, for

granting the author leave of absence during this investigation.

The work was supported by the Defence Research Board of

SUMMARY

Measurements of the turbulent stresses in fully-developed

pipe flow have been made as an overall test of the OISA

constant-temperature hot-wire anemometer. The measurements were made with

single slanting wires and a normal wire.

The shearing stress was measured at two Reynolds numbers

and compared with values computed from the pressure drop down the

pipe: an accuracy of 10% or better was achieved. The computed

longitudinal and transverse normal stresses were in good agreement

with the measurements of Laufer. It was confirmed that the heat

loss from the wire varied linearly with

uO.45

(collis' law)

0.50 ( , )

CONTENTS Page NOTATION 1. INTRODUCTION 2. THEORETICAL ANALYSIS

3.

EXPERIMENTAL INVESTlGATIONS

4.

DISCUSSION OF RESULTS

5.

CONCLUSIONS REFERENCES iv 1 2

6

9 12

13

NOTATION

a, band c constants in the law describing the cooling of the wire (equation

(1))

d diameter of pipe

e instantaneous bridge voltage fluctuation

atmospheric pressure

static pressure

pitot total pressure

R operating resistance of hot wire

r radius of pipe

wire resistance at still air temperature

Reynolds number (= UCd)

v

.

Tw bot.-w.Lre operating t.emperature

Too ambient temperature

Ulocal mean flow velocity

u longitudinal fluctuating component of velocity

Uc mean flow velocity at centre of pipe

=

("pol~

U,. skin friction velocity

V br1dge O.C. voltage

v transverse fluctuating component of velocity

Vo bridge O.C. voltage at zero air speed

x d i.st.ance along the axis of pipe

y distance from wallof pipe

a function of bridge O.C. voltage (= 2V )

v2_v~

v kinematic viscosity

p density

'r turbulent shear stress

₌

-puv

'ro shear stress at y

=

0

1 . INTRODUCTION

Fo11owing the pioneering work of King(l) on the convection

of heat from sma11 cylinders in a stream of f1uid, much use has been made of hot wires in the measurement of turbulence. The hot-wire turbu1ence-measuring techniques have now been developed to a stage that the hot-wire anemometer has become a useful tooI in the routine investigation of turbulent shear flows. The most recent review (1955) of hot-wire techniques has been given by Cooper and TUlin. (2)

The purpose of the present investigation was to measure the Reynolds stresses in fully developed turbulent pipe flow with a single slanting wire (for shearing stress and transverse

turbulence) and anormal wire (for longitudinàl turbulence) using a constant temperature anemometer(4) (DISA. 55A01) and thereby to obtain an overall check of the instrument.

The resu1ts presented in this paper are for two single slanting hot-wires at pipe Reynolds numbers of 2.74 x 105 and

3.64

x 105• The measurements of normal Reynolds stress were made at the ~ower Reynolds number using anormal wire, and the results

2. THEORETICAL ANALYSIS

Assuming that the convective heat loss from the wire

varies linearly with UC, where U is the instantaneous velocity

of the flow and c is a constant, the working equations for a

constant temperature anemometer can be readily derived.

In the DISA

55AOI

anemometer(4) the hot wire forms one

arm of a bridge circuit as shown in Fig.l. The wire is maintained

at constant temperature by means of the feed-back circuit to the

bridge. The mean bridge voltage is measured and, being proportional

to the wire current, is related to the mean velocity of the flow.

Across the bridge terminals an additional circuit i~

connected to measure the bridge voltage fluctuations which

correspond to the flow fluctuations or turbulence.

Assuming that the wire is cooled only by that component

of the flow velocity which is perpendicular to the hot-wire(6)

then, for a wire yawed to the mean flow at an angle ~, the

governing equation can be written:

V2_

R _ a + b(U (R - Ra) ....•... (1)

where V is the bridge D.C. voltage,

R is the hot-wire operating resistance,

Ra is the wire resistance at still air temperature,

a and bare constants depending slightly on the

. (6) (8)

w~re temperature ,

U is the mean flow velocity, and

Equation (1) was originally proposed by King with

c

=

0.5 .

.

However later and more accurate measurements by Collis indicated that c

=

0.45

for the usual range of wire Reynolds numbers (less than

44).

For fluctuations in velocity about the mean, equation

(1)gives for constant R,

2V c-l 1

R(R-Ra) dV

=

b c cost sin t UCdt + b sinc~ cUc- dU • .. (2) If the fluctuating components u and vare small compared with mean velocity U

dU c u and Ud~

=

v · .. (3)

Substituting equations (3) in equation (2), 2V dV

=

b sinCt Uc-l [cu + "IvJ

R(R-Ra) • •• ( 4 )

where _'Y

₌

c cot ~ _{· .. (5)}

If Vo be the value of V corresponding to U

=

0 in equation

(1)

a

=

R(R-Ra) _{· •• (6)}

Substituting equation (6) in equation

(1)

,

b sinc~ UC

=

1 (V2 _ v2)

R(R-Ra) 0 • •. (7)

Equation (7) in equation (4) gives:

• .. (8)

or

_t3

_e

₌

_ti1_:(cu

₊

_{"Iv) • .. (9)}

in bridge voltage.

squaring equation (9) and taking the time mean,

2 2 1 (2 -2 2

"2

-)

~ e

=

U2 c u + ~ v + 2~c uv · .• (10)

2

Taking two readings of e by re-ori.entating the single slanting

wire in the plane defined by the wire and the mean velocity U,

from an angle of yaw of + ~ to an angle of yaw of - ~, equati.on(10)

becomes: ~2

₁

,e2

₁

= -2

1

(c2 u2 + ~2 ~ + 2,,/cuv) U

·•.(11)

2 e2

1

(c2 u2 2 2 - 2,,/c~) ~2 ₂ = U2 + "/ v

where ~l should equal ~2 if the wire is correctly orientated with respect to the mean velocity.

Subtracting equations (11):

-puv

= 1.

2c 1 "/ 1 u2 2 P • • 0 ( 12)

Thus the turbulent shearing stress can be measured with a single slanting wire in conjunction with the constant-temperature

anemometer (DISA.

55A01)

provided the constants c and ~ are

known. If the calibration curve for a hot-wire follows King's

law then c

= 0.5

and ~ would then be

0.5

cot~ (equation

(5».

Newman and Leary(3) have used c

= 0.5

and "/

= 0.457

cot~ for

a constant current anemometer and obtained goed agreement in

COllis(8) has noted that this is consistent a similar pipe test.

line for constant current operation due to the small variation

of a and b with temperature. However for constant-temperature

operation c

=

0.45 and ~

=

0.45 cot~. The latter values are

used in the present paper.

It is also interesting to note that equation (10)

is directly applicable to a normal wire for which ~

=

o .

=

. . . (13)

The addition of equations (11) for a slanting wire, in

conjunction with the normal-wire readings,gives an equation

v2

turbulence ~ may be computed.

U

from which the transverse

It is interesting to note that the assumption of King's

law rather than Collis' law would lead to values of the computed turbulence stresses which were 19% too low.

3

.

EXPERIMENTAL INVESTlGATION

The general layout of the experimental apparatus is

shown in Fig. 3. The apparatus was primarily designed to study

radial channel flow(9) however for the present investigations

the radial channel section at exit of the pipe was removed. The

pipe consisted of

3

ins. I.O. precision brass tube approximately

14 feet in length. Statie pressure taps (0.015 ins. diameter)

were provided at various locations along the length of the pipe.

The hot-wire traverses were made about 2 ins. upstream of the

p.ipeexit. A double ended dial gauge with 0.001 inch graduations

was used for traversing the flow.

Air was supplied from a centrifugal compressor, driven

by a 10 H.P. constant-speed, three-phase motor. A fibre glass

filter (approximately 0.002 ins. fibres) was provided at the

compressor entry to reduce dust accumulat.i.onon the hot.-w.:r.e

and this was found to be extremely effective. The mass flow

in the pipe was roughly controlled by a bleed valve situated

at the compressor outlet.

The compressor was situated about 40 feet away from

the supply station .inthe Aerodynamics Laboratory and thus the

supply pipe was sufficiently long to damp out any large

fluctua-tions emanating from the compressor. The bends in the supply

pipe were gradual. The precision brass pipe was connected to

the supply station by 12 feet of flexible tubing. An assembly

-

7

-at the junction of the flexible tubing and the brass pipe. The

deep cell honeycomb was used to straighten the flow and the

symrnetricalbleed valve was used for fine speed control.

The hot-wire anemometer used for the present investiga-tion was a commercial unit supplied by DrSA Electronik

Als

of Denmark. The operating procedures for this unit is described

in their instruction manual(4).

The hot wires were made from platinum-coated tungsten of nominal diameter 0.0002 ins. They were operated at a

resistance R

=

1.8 Ra' corresponding to a temperature about 2000C above arnbient. The measurements were made using two

single s1anting wires (to check repeatabi1ity) and one norma1

Ucd

5

wire. Tests were made for two pipe Reynolds numbers

=

2.75xlO v

and

3.6

4

x 105.

The angle of yaw

(

t

)

between the wire and the axis of rotation for each slanting-wire probe was determined by mounting the probe in a photographic en1arger with a magnification of about 10.

The wires were examined under a microscope before and after each test to check for any accumulation of dust. The fibre glass filter at the compressor inlet was effective in preventing this for the present tests.

Correct alignment of the slanting wire probe with the axis of the pipe was assured when the mean voltage reading was independent of the rotation of the wire.

Pitot traverses of the pipe were made with a 0.030 ins.

O.O. pitot tube with sharpened lips. Wall statie pressures were

4. DISCUSSION OF RESULTS

Verifieation of Collis' Law:

It was eonsidered desirable to eonfirrnCollis' law

before rnakingany turbulenee measurements. The normal wire

was therefore mounted at the eentre of pipe and velocity there

varied from

°

to approximately 336 ft./see. The round pitot

tube was used to measure this velocity, the statie temperature

being taken as ambient. Sinee the hot-wire readings are sensitive

to the temperature of the air stream and this varied slightly, it

was neeessary to apply corrections. To do this Collis'

was assumed, the variations of the therrnaleonduetivity and the

kinematic viseosity of the air with the ambient temperature

being negleeted. Taking logarithmie differentials:

6V

₌

6T-:o_{[ 0.17 Q.:.ll}

Tw=Too

J

V _{2 Tw+Too TM •.. (14 )}

gives the required correction to the bridge voltage.

The statie temperature of the air was measured with a

mereury thermometer and, where neeessary, compressibility

eor-rections were applied assuming a recovery factor of 0.90.

Pig.(2) shows the measured values of

u

O

.

45

plotted

against V2 the latter being eorreeted to an ambient temperature

of 90oP. In all cases the corrections to bridge voltage were

less that l~%. The linearity of the results substantially

eonfirms Collis' law. Purthermore the zero-wind readings lies

may be taken as the measured bridge voltage with'wind off.

Fig.(4) shows the pressure drop along the pipe for the

two Reynolds numbers of 2.74 x 105 and 3.64 x 105 and is seen

to be satisfactorily 1inear. The slope gives the distribution

of shearing stress across the pipe(3) and is shown in Figs. (7)

and (9).

Pitot Traverses:

From equation (12) it is seen that the computation of turbulent shearing stress, -püV, requires a knowledge of

~ pu2• This information could be obtained from the mean hot-wire readings. However in this case the variation of ~ pU2 was determined more directly by traversing the flow with the round pitot tube. Fig.(5) shows the measured values of dynamic pressure across half the pipe at the section where hot-wire measurements were taken.

Shearinq Stress Distribution:

Fig.(6) shows the single slanting wire readings for two different orientations. The measurements were made at Reynolds number of 2.74 x 105. The single slanting wire had a

value of ~ = 430 - 40'. From the smoothed curves of Fig.(6) together with Fig.(5), the value of

-püV

was computed. This is shown in Fig.(7). In this figure the straight line shear-stress distribution obtained from the pressure drop is also p10tted for

comparison. The shear stress variation obtained from the single

s1anting wire is in good agreement with that obtained trom the

pressure drop.

To establish further confidence in this method an

alternative probe (~

=

450)was chosen to traverse the flow at

the higher Reynolds number of

3.6

4

x 105. The measurements with

this probe are presented in Figs.(8) and

(9).

Once again it is

seen that the hot-wire measurements are in substantial agreement with those obtained from the pressure drop.

Longitudina1 and Lateral Turbulence Measurements:

Anormal wire was used to measure the longitudinal

turbulenee in the pipe at Re

=

2.74 x 105. In Fig.(lO) the

longitudinal turbu1enee made non-dimensional in terms of the

u2 v

U~' is plotted against

(?).

Similar

by Laufer(5) at a Reynolds number of

sk.tnfriction velocity

measurements were made

5.0 x 105 and these are also shown for eomparison. The agreement

is good.

v2

The transverse turbulenee was eomRuted from both

U~

the s1anting and normal wire readings, and is shown in eomparison

with Laufer's measurements in Fig.(ll). Onee again the

CONCLUSIONS

The measurement of the Reynolds stresses in fully

developed turbulent flow down a circular pipe is commended as

a reliable means of checking a hot-wire anemometer. It is

demonstrated that, for the level of turbulence encountered in

a pipe, the turbulent shearing stress can be measured with a

single slanting wire controlled by the DlSA anemometer to an

accuracy within 10%. Furthermore the measurements of langitudinal

and transverse turbulence are in good agreement with those of

Lau~er(5) .

It is noted that Collis' law rather than the conventional

King's law must be used to determine the turbulent stresses when

1. King, L.V. 2. Cooper, R.D. Tulin, M.P. 3. Newman, B.G. Leary, B.G. 4. DISA Manual 5. Laufer, J~ 6. Collis , D.C. Williams, M.J. 7. Schubauer, G.B. Klebanoff, P.S.

8.

Collis, D.C. 9. Molier, P.S. REFERENCES

On the Convection of Heat from Small Cylinders in a Stream of Fluid.

Determination of Convection Constants of

Small Platinum Wires with ·X~lication to

Hot-Wire Anemometry.

Phil. Trans. Roy. Soc. (London) Ser.A:

Vol 214, pp. 373-432. 1914.

Turbulence Measurements with the Hot-Wire Anemometer.

AGARDograph 12. August 1955.

t

The Measurement of the Reynolds Stresses in a Circular Pipe as Means of Testing a Hot-Wire Anemometer.

Aeronautical Research Laboratories, Report A.72, Australia, November 1950. DISA Constant Temperature Anemometer 55A01 Instruction Manual.

DlSA Elektronik

Als ,

Herlev , Derimaxk,1962.

The Structure of Turbulence in Fully Developed Pipe Flow.

N.A.C.A. TN.2954, June 1953.

Two-Dimensional Convection from Heated

Wires at Low Reynolds Numbers. J.F.M., Vol.6, pp.357, 1959.

Investigation of Separation of the

Turbulent Boundary Layer.

N.A.C.A. T.N.2133 August 1950

•

Forced Convection of Heat from Cylinder

at Low Reynolds Numbers.

J. Aero. Sciences, Vol.23, pp.697,

July 1956.

Radial Flow Without Swirl Between

Parallel Disks.

M.Eng. Thesis, Dept. of Mechanical

PUSH TO MEASURE r-

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i ! ' t::. Re

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PITOT TRAVERSE ACROSS HALF

THE PIPE

,

~++~~·~~~,+-t~~~~+1~~'_~+1-r~'_~+1-r~'_~~-r++,

I

FIG.

8

SLANTING WIRE READINGS ACROSS HALF THE PIPE

SHEAR STRESS DISTRIBUTION ACROSS HALF THE PIPE

, I

I

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LONGITUDINAL TURBULENCE

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