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Interpretation of a hot wire signal using a universal calibration law
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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.
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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
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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
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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.
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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
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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