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Interpretation of a hot wire signal using a universal calibration law

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1971 J. Phys. E: Sci. Instrum. 4 225

(http://iopscience.iop.org/0022-3735/4/3/016)

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2/8

Interpretation of a hot wire signal

us ing a universal calibration law

H H Bruun

Institute of Sound a nd Vibration R esearch, University of

Southampton, Southampton

SO9

5 N H

MS received 2 October 1970

Abstract This paper describes the interpretation of the ho t

wire signal from a single hot wire in terms of a universal

function. Th e calibration curves for probe s with nom inally

the same geometry have been studied for deviations in the

shape of calibration curves, yaw dependence an d the effect

of angle of incidence. Fr om these results the basic equations

for the interpretation of the signal from n orma l and yawed

hot wires have been derived.

1 Introduction

The interpretation of hot wire measurements of the turbulent

velocity components depends on a detailed knowledge of the

steady heat transfer fr om electrically heated cylinders. Investi-

gation of the heat transfer process fo r flow varying from free

molecular to continuum and speeds from those of natural

convection to supersonic have shown tha t the Nusselt number

for an electrically heated wire in general is a function of

several parameters :

(1)

where

CL

is the angle between the velocity vector an d the norm al

to the wire.

This paper describes the use of hot wires in air flows at

atmospheric pressure (windtunnels, air jet flows, etc.). Th e

Prandtl number ( P r ) is therefore constant. The effect of the

Grashof number, as shown by Collis and Williams

(1959),

will only be significant at extremely low velocities, permitting

this parameter to be omitted in

most

practical anemometer

applications.

The maximum velocity in this investigation was limited to

150m s-1, and only a given hot w ire type with know n nom inal

values of Lld and of the overheat ratio TWITwas used. These

restrictions reduce the influence of the Mach number and of

the Knu dsen num ber to second orde r effects.

The flow temperature during all the experiments was kept

constant and equal to the room temperature. It is therefore

possible to relate the Nusselt number to the square of the

voltage output E 2 and the Reynolds number to

pV,

giving the

following heat transfer relationship

:

(2)

Nu=

F (R e , Gr, Pr, Kn, M,

Lid, TWIT,

E 2 = K (pV, CY, L ld , Tw/T).

To date it has been customary to express the relationship

(2) in the form

(3)

where

V ,

is the effective cooling velocity, often set equal to

the normal component of the velocity, i.e. Ve=Vcos

E

The use of this assumption may lead to a considerable error,

as shown in

03.

King

(1914)

gave the value

of 0.5

of the

exponent

n,

while Collis and Williams

(1959)

found

n=0.45

fitted their data better. The constants A and B are norm ally

E*

=

A +B p

V,).

determined experimentally, with A either as the heat loss at

zero flow speed or as the intersection value of the E2 axis.

Relating equation (3) to measured hot wire calibration

curves has revealed that A and B cannot be assumed constant

over a large velocity range. Kings or Collis and Williams

law are therefore only approximations applicable over

a

limited flow range. This aspect is discussed in detail in 2.3.

2 A universal empirical heat transfer law

2.1 Hot wires normal to low direction

The form of equation (2) suggests that for a given hot wire

probe type it may be possible to describe the calibration

curves in terms of a universal heat transfer law, thereby

reducing the necessary calibration work considerably.

The hot wire probe type used in the investigation was t he

2

mm ISVR h ot wire probe illustrated in figure

1

The sensing

element of this probe type consists of a 5 p m tungsten wire.

Figure

1

The

2

mm ISVR probe

By

a

copper plating procedure the active length of the sensing

element has been removed from the prongs (Davies an d D avis

1966).

The active length obtained by this procedure was foun d

to vary between

1.8

mm and

2.1

mm. The ratio of the total

length of the wire and the active length is approximately

2.5

(see figure

1).

The prongs are

8

mm long. D ue to variation of

the active length, the diameter of the wire and the resistivity

of the tungsten m aterial, these probes ha d a scatter in the cold

resistance of the order of

5-10 .

The hot resistance was set

to a constant value of 15

2

giving a nearly constant overheat

ratio Tw/Tof

2.

Calibration measurements with hot wire probes having the

probe support placed perpendicular to the mean flow were

carried out in an open circuit wind tunnel and in a

2

in air

jet having a stagnation temperature equal to the room

temperature and expanding into th e atmosphere.

225


3/8

H H

Bruun

In describing the calibration law the heat transfer was

expressed in terms of

E*- Eo2

as a function of pV.

A

small

container having a diameter of 1.5 cm was used to shield the

hot wire for the measurements of the voltage output

EO

a t

zero flow speed. The measured calibration curves indicated

that it was possible to express the heat transfer law for this

hot wire type in terms of a universal function

f ( p V ) .

The

calibration law was therefore expressed as

E2-Eo2=

C f ( p V ) (4)

where

C

is a constant which must be determined individually

for each wire.

A

universal function

f V )

as been calculated

for air jet flow from the calibration curves. The function

f ( V )

s given in table 1 in terms of

E.

The density has been

omitted as a parameter for this flow type as

p r)

s identical

for all such calibrations.

Two additional types of measurements were carried out to

check the hypothesis of

a

universal shape of the calibration

curves. In the first type of experiment two 2 m m ho t wire

probes

A

and B were placed simultaneously in the centre of

a 2 in air jet, having the supports perpendicular to the mean

flow. If the hypothesis of a universal shape is valid then the

ratio

(E2 Eo2)a/(E2-E o ~ ) B

hould be independent of

Gx

r f

I

80

- .10,

I I

20 40 60

Velocity ( m s- )

Figure 2

Difference in shape of calibration law for two sets

of two 2 mm long hot wires

velocity. Two typical results of such sets of measurements

have been plotted in figure

2. A

small variation in the ratio

(E 2- Eo2)a/(E2-E o ~ ) B

ith velocity is observed. This is due

to a slight difference in the shape of the calibration curves of

hot wire

A

and B. The uncertainty in the estimated velocity

caused by this small variation in shape was calculated for

several calibrations. Fo r velocities above

10

m s-l the uncer-

tainty was evaluated to be of the or der

f

. Below

10

m s-l

the uncertainty increases slightly with decreasing velocity a nd

becomes of the order k 2 to 3 at

1

m s-l. However, if the

maximum velocity of the calibration curve is lowered then the

uncertainty in the lower velocity range is reduced corre-

spondingly.

Due to small imperfections in some wires (dust accumula-

tion, etc.) drift could not always

be completely avoided.

Measurement of

EO

before and after the calibration run gave

the magnitude of the drift. The observed error introduced by

drift was found to be quite consistent with the calculated

additional error in the estimated velocity. The measured

calibration curves showed that a drift in

EO

f

2

5 mV gave

an additional uncertainty in the velocity of f 1 . Similarly,

an evaluation based on the calibration curve

f ( V )

predicts

that a 1 change in

V

corresponded to a change in

E

of

3-5

mV in the whole velocity range. The additional error

introduced by small amounts of drift can therefore readily

be explained. However, as the source and time of the occur-

rence of the drift is usually unknown, this erro r can norm ally

not be compensated for.

The accuracy of the chosen standard fun ction f( V ) table

1)

was also investigated. Several calibration curves with different

2 mm hot w ires placed one at a time in a

2

in air jet was

measured in terms of

E* EO*

s a function of V. Th e velocity

was determined by accurate man ometer readings. Th e velocity

was then recalculated using the standard function

f V )

and

the assumption of a constant ratio

( E 2 - E02)/f(

) . After

some initial corrections to

f (

V ) incorporated in table

1)

the

difference between the velocity calculated by the se two

methods could be explained by the total error of the small

difference in shape (error

k

+ ) and the velocity uncertainty

(error f + ) from the manometer reading.

These measurements have justified the use of a universal

shape of the calibration curves for hot wires with the same

nominal geom etry. Fo r velocities greater than 10 m s-l the

error in the estimated velocity introduced by this simplifica-

tion will be less than

c 1 .

Below

10

m

s-1

the uncertainty

increases slightly with decreasing velocity, being of the order

of

i: 3

at

1

m

s-1. If

the hot wire voltage drifts during the

experiment the uncertainty will increase with

1%

for each

5 mV drift.

The above measurements were carried out for a constant

probe geometry.

A

few tests were carried out to investigate

whether the concept of a universal shape is applicable to

probes with different geometries. ISVR hot wire probes having

active lengths of

1

mm and 3 mm and run at an overheat

ratio of 2 were used for this purpose. First a

1

mm and a

2 mm hot wire probe was placed in the centre of

a

2 in air

jet with their supports perpendicular to the main flow. The

ratio

(E2- Eo~ )I/(E *E092

was recorded as function of the

velocity

V,

using the 2 m m hot wire probe for the velocity

determination. The same measurements were then performed

with a 2 mm and a 3 mm hot wire probe. Both sets of results

showed the sam e type of curve variation. Only the results for

the 2 mm and 3 mm hot wires have therefore been presented

in figure 3. The figure shows a somewhat greater difference in

shape than in figure

2.

The variation is, however, only 2-3

times greater than in figure 2, giving

8

difference in velocity

estimate at 1 m s-1 if the same universal function f ( V ) is

used. Th e similar trend of the two sets of experime nts, however,

indicate that a different universal function can be used for

other types of hot wires.

Changing the support orientation relative to the mean flow

also has a minor effect on the calibration curve. This is

described in

4.

2.2

Analytic approximation

of

the calibration law

In evaluating turbulent data the calibration curve f p

V )

s

often approximated by an analytic expression. To obtain

accurate results, two conditions must be satisfied. First, the

mean voltage

E

and the mean flow value must satisfy

the analytic expression, and secondly, the slope variation of

Figure

3 Difference in shape of calibration law for two sets

of 2 mm and 3 mm long hot wires

226


4/8

Interpretation of a hot wive signal using a unicersal calibration law

the analytic expression must be the same as the calibration

curvef(pV ) around the point

(E, pV .

A convenient way of

expressing

f ( p V )

s the power law

Kl(pV )nl

giving

- -

E'- Eo2=K1(pV).l

(5)

where

KI

and

nl

are functions of the velocity.

In many hot wire measurements only the velocity fluctua-

tions are of interest. By including the mean density variation

with velocity in the variable

K,

equation

(5)

can be rewritten as

E'- Eo2=KVn.

(6)

The change in

n

with the velocity for air jet flows is shown

in figure 4. The evaluation of

n

was carried out by the proced-

ures given in the appendix using the fun ction

f

( V ) (table 1).

I

1

, ( I I

IO

0,301

I S I I

I

100

Velocity (m s-l)

Figure 4 Variations of exponent n with velocity

This

way of specifying n V ) nsures that the above mentioned

slope requirement is satisfied. The investigation by Kjellstrom

and Hedberg (1968) with 1 mm DISA hot wire probes gave

a similar variation

of

nl with pV. Their data evaluation was

based on method (i) in the Appendix and shows some vari-

ation in n1 with prong configuration.

Fo r practical hot w ire applications it is necessary to assume

that the variable K and

n

in equation (6) are constants.

Knowing the mean velocity of interest V, the corresponding

value of n is selected from table 1. The value of K is then

determined from the measurements of

EO

nd one accurate

measurement

of

corresponding values of

E

and

V.

By choosing

these values of n and K the approxim ation of a constant value

of K and n can be applied over a considerable velocity range.

Two different criteria were used for determining this velocity

range. In the first the limits of the velocity range w ere deter-

mined by a set maximum deviation ( *

X )

in the estimated

velocity, in the following denoted as the

A

velocity range.

Values

of

1 and 5 % were used for X . This criterion deter-

mines the maximum error in the mean velocity determination

due to curve approximation. As the second criterion a set

maximum deviation

(+

Y ) in the estimated value of the

slope dE/ dV of the c alibration curve was used, in the following

denoted as the

Y

slope range. A value of 5 was used for

Y.

This criterion determines the maximum error in the fluc-

tuating signal interpretation due to curve approximation.

The velocity ranges for values of the mean velocity

P

going from 1 m s-1 to 150m s-1 has been calculated by using

the function f

( V) .

Th e results ar e plotted in figure 5 i n terms

of

the ratio

Vmax/Vmin.

Only points up to

l o o m s - l

have

been plotted, as

Vmax

exceeds 150 m

s-l

for mean velocities

above

100ms-1.

Curves A and B correspond to the 1 %

and the

5

velocity range while curve

C

corresponds to the

5 % slope range. Th e range corresponding to the 5% uncer-

tainty in the slope (curve

C )

s seen to be similar to the 1

velocity range (curve A) above 30-40 m

s-l.

Below 30 m s-l,

Table

1

Universal calibration functionf( V ) or a

2

ot

wire probe operated at 15 Ll

Air temperature 18C

Velocity V Output

E

Exponent

n

Exponent

m

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1*6

1.8

2

2.5

3

3.5

4

5

6

7

8

9

10

12

14

16

18

20

25

30

35

40

45

50

60

70

80

90

100

110

120

130

140

150

1,167

1.269

1.336

1.387

1.425

1.457

1.486

1.511

1.533

1,554

1,570

1.611

1.648

1.681

1,711

1,764

1.810

1.852

1.889

1,923

1.955

2.012

2.062

2.108

2.149

2.187

2.270

2.341

2,405

2.458

2.509

2.553

2.632

2.702

2.761

2.814

2.862

2.906

2,946

2.982

3.017

3,049

(eqn 15)

0.72

0.65

0.60

0.56

0.54

0.53

0.52

0.52

0.515

0,510

0.505

0,505

0300

0.500

0.495

0.490

0,490

0.485

0.485

0.480

0.480

0.475

0.470

0.465

0.460

0.450

0,445

0.440

0,430

0.425

0.415

0.405

0.400

0,390

0.385

0,375

0,370

0.365

0.360

0,360

0.51

0.50

0.50

0.49

0.49

0.48

0.48

0.465

0.460

0.455

0.445

0.440

0.435

0,430

0,425

0.420

0.415

0.415

0.410

0.405

0.400

0,395

0.390

0,385

0.380

0,380

0.375

0.375

curve

C

is, on average, midway between the

1

and 5%

velocity range. T he great increase in

VmaxlVmin

at the lower

velocities is, however, mainly due t o a small change in

Vmin,

giving large values of

Vmax/Vmin

at low velocities. The 1

velocity range and the 5 slope range were therefore found

to be nearly equal in terms of the width of th e velocity range

which can be defined as +(

Vmax Vmin).

For fu rther compari-

son the

1

velocity range was chosen. This range is described

in mo re detail in table 2. By assuming a G aussian probability

function of the turbulent fluctuations it is possible to relate

the velocity range

to

turbulence intensity. By setting

Vmax-

V

equal to 3a, where is the stan dard deviation, and similarly

setting V -

Vmin

equal to 3a and averaging the two results,

the turbulence intensities given in table 2 were obtained. At

227


5/8

H H Bruun

A

I

O h

v e l o c i t y r a n g e

5 10 v e l o c i t y ra n g e

\

1

20 40

60 80

100

V e l o c i t y ( m

s-1

Figure

5 Velocity dependence of the ratio Vmax/Vmin

low flow velocities the turbulent intensity has further been

restricted

so

that no negative velocities will occur. This table

shows that above

30-40

m

s-1

the approximations of constant

K

and n can only be applied to flows with less than

15

turbulence intensities if less than 5 % uncertainty is to be

introduced due to curve approximation. Fo r higher turbulence

intensities more complicated expressions for the calibration

curve must be used to overcome this problem. The use of

Collis and Williams' and King's law in this region will, as

shown in

52.3,

give an even worse approximation.

Below 30 m

s-1

the approximation of constants

K

and n

becomes considerably better, permitting flow with up to

30

turbulence intensity to be studied with less than

5 %

uncertainty due to curve approximation.

Changing the exponent

n

in equation

(6)

to

a

value different

from the value corresponding to the mean velocity 7 was

Table 2 Velocity ranges corresponding to maximum

1

uncertainty in the velocity estimate using the calibration law

equation (6)

1

2

4

6

8

10

20

30

40

50

60

70

80

90

100

105

1.5 0.7

5.0

0.9

10

1 5

16

2.5

18 3.5

22

5.0

34 9.6

48 16

65 26

70

30

85

36

95

46

105 54

120 60

130

70

150

85

2.2 15

5.6 30

t

6.7 40 f

6.4 351.

5.1

3o.t

4.4

301.

3.8

25

3.0 18

2.5

15

2.4 14

2 . 4

13

2.1 12

2.0 11

2.0

11

1 9 11

1 8 10

+ Reverse flow will occur at higher turbulence intensity

found to reduce the velocity range considerably. By adding

0.05 to the value of the exponent

n,

the

1%

velocity ran ge

was reduced from

2.5

to

1.2

for velocities above

3 0

m

s-l.

At the same time the requirement of matching slope of

approximation and calibration curve was no longer satisfied.

2.3

Comparison with King's and Collis and Williams' law

The usual way of expressing the calibration law is

( 7 )

2= A + BVn

usually known as King's law for

n=0.5

and as Collis and

Williams' law for

n= 0.45.

These laws too have been compared with the calibration

law

(6)

by calculating their

1

velocity ranges. By differenti-

ating equation ( 7 ) one obtains

B=2E(dE/dV)/ (

Vn-ln).

(8)

Inserting the values of n equal to

0.5

and

0.45.

and the values

of E and dE /dV from the function

f V)

he B variations

shown in figure

6

were obtained. T he corresponding values

of

A

were obtained from equation

( 7 )

and are also seen plotted

in figure

6.

The value of

A

and B in Collis and Williams'

law below 20 ni s-l is seen to be nearly constant, indicating

a

very good approximation uhen

A

and

B

are assumed

constant. Evaluation of the

1

velocity range for King's

law and Collis and Williams' law was carried out using the

values

of

A and B given in figure

6.

The results a re given in

table

3.

By comparing the velocity ranges with the results in

table 2, King's law is seen to be slightly inferior to e quation (6)

at a ll velocities.

Collis and Williams' law is slightly inferior at higher

velocities and is a better fit below 20 m

s-1.

In this velocity

region, as pointed out in 52.2, equation

(6)

can be applied

to Bows with turbulence intensities above 30 without

introducing more than maximum

5

error due to curve

approximation. The extension

to

higher turbulence intensities

by using Collis and Williams' law can only be achieved by

the determination of A( V) and B ( V ) for each individual

wire. The use of any other value of B than the value given by

equation

( 7 )

will immediately reduce the applicable velocity

range.

These calculations show th at only below 20 m s-l is Collis

and Williams' law a better approx imation th an the calibration

law

(6).

This law is, however, such a good fit in this region

that the normal extra calibration procedure necessary for

using Collis and Williams' law seldom can be justified.

228


6/8

Znterpretation of a ho t wire signal using a uniseusal calibration law

V eloc i t y

( n

s- )

A 5

Table

3

Velocity ranges corresponding to maximum 1

uncertainty in the velocity estimate using the calibration

law

7)

cy

y

5

0 . 9 0 -

3

0 . 8 5

King's law

Collis and Williams' law

o 24

. x

6 ~

x x

x

x X L X

* x

A

A X X

r O X

0

A o O

I I

I

I I

01

1 1.5

0.7 2.2 2.0

0.7

2.9

2 4.5

1.0 4.5

10

0.8

12.5

4 9.0

1.5 6.0

30

1.0 30

6

12 2.5

4.8

26

2.0 13

8 16 4.0

4.0 26

2.0 13

10

18 5.0 3.6

26

2.0 13

20 32 12

2-8 36

9.0 4.0

30

46 20

2.3 48

16

2.9

40

65

30

2.2

70

30

2.3

50 70 34

2.1

75 34

2.2

60

85 44

1.9 85

42 2.0

70 95 52 1

*8 100

50 2.0

80

110 60 1

8

110

60 1.9

90 120 65

1.8 125

65 1*9

100 135 75

1.8 135

75 1a8

The universal law (4) requires only one accurate measure-

ment of corresponding E and V value as well as the m easure-

ment of E O, ut a small error due to sm all variations in shape

is introduced by this method'(see $2.1). King's law and Collis

and Williams' law has the advantage of permitting an indi-

vidual calibration of each wire. However, unless

A ( V )

and

B ( V ) are determined for each test run, these laws will not

give any improved accuracy above the universal shape

approach.

3

Hot wires yawed to flow direction

The heat transfer from hot wires yawed to flow direction, in

the following denoted as 'yawed wires', has been studied by

several investigators (Kjellstrom a nd Hedberg 1968, Sanborn

and Lawrence 1955, Webster 1962, Champagne e t al. 1967

Friehe and Schwarz 1968).

A

summary of these findings has

been given by Bruun (1969).

The angle a between the flow direction a nd the norm al to

the wire is in the following used to describe the positioning

of the wire.

Introducing the effective cooling velocity

Ve

=

Vf

a)

(9)

in the calibration law equation (7), gives

E,'=A+BVen.

(10)

In many hot wire measurements normal component cooling

or cosine law cooling, i.e.

Ve=

Vcos

a,

has been assumed.

The above mentioned investigations, however, have shown

that a considerable error can be introduced with this assump-

tion.

To correct for this deviation, several expressions for the

function

f ( a )

have been suggested, normally applicable for

a e

60 . Some of these are, however, too cumbersome for

practical applications.

Two applicable expressions for

f a )

are

f1(a)

= co s

m q a )

(1 1)

f2 a) =

(cos

+ k2

sin

2a)l; .

(12)

The yaw parameters

k

and m1 will, in general, be functions

of

a

and V and to

a

minor extent of Lid, TWIT,wire material

and prong configuration.

3.1

Experimental investigation of yaw param eters

The heat transfer measurements for the yawed wires were

performed in the potential core of a 2 in air jet. Th e velocity

range investigated was

1

m

s-1

to 60 m s-1. Th e velocity was

determined with a 2 m m ho t wire probe using the universal

calibration law (equation (4)). The yawed hot wire probe

was placed horizontally, having the support perpendicular to

the flow velocity. The measurements of

01

were performed

with a telescope fitted with a graticule and mounted on the

opposite side of the jet fro m the probe. In this way

a

could

be determined to within

f + .

The yaw parameters

k

and ml were determined from the

universal calibration law in the form

Ea2

Eo2

=

Cf e).

(1 3)

E, was measured as a function of the angle

a

and of the

velocity

V.

By measuring

EO

and

Ea=o

as

a

function of

V,

the constant C in equation (13) could be calculated, Knowing

these quantities, e quation (13) was then used

to

calculate

Ve

as a function of

V

and

a.

By using equations

(lo), (11)

and

(12) the dependence of the yaw parameters

k

an d ml o n

a

and

V

was evaluated.

The result for the yaw parameter

k

is plotted in figure 7

for three different velocities, showing

a

large variation in k

with

a

for this probe type and probe support orientation.

The dependence of the yaw parameter ml on

a

is plotted in

figure 8 showing ml to be nearly independent of

a

for

a

< 70 .

The velocity dependence of this yaw parameter is seen from

figure 9 to be very small.

0 . 3

n

0.

X

X

A

(m 5-11

*

I I

I I I I

20

40

60 80

A n g l e o f

y a w

Figure 7

Dependence of

k

on angle of yaw a nd velocity

Angle o f y a w

Figure 8

Dependence of ml on the angle of yaw

229


7/8

H

H

Bruun

I

I

I

I

0 IO

20 30

40

50

60

Velocity (m s-1)

Figure

9 Variation in m1 with velocity

For practical yaw measurements it is necessary to express

equat ion

(13)

in an analytic form. Using the power law the

equation takes the form

where K and

n

are constant values corresponding to

Ve=

V(x =O 0). This law is, as mentioned in

$2.2,

only an

approximation to equation (13). For large values of

V/Ve

(i.e. a large), deviations between Ve calculated from equation

(13) and equation (14) will occur and therefore also deviations

in the yaw parameters. The difference in the yaw parameters

calculated from

(13)

and

14)

was negligible at sm all values of

a.

At 45 the difference amounted to approximately 5 % and

a t 60 to 10% . Remembering that the yaw parameters only

are corrections to the cosine law, this uncertainty in the yaw

parameter will only amount to

1-2%

relative uncertairdy in

the magnitude of the turbulent quantities.

The yaw parameter ml is seen to be nearly independent of

a

and V

( a

bruun hot wire

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