J. Phys. E: Sci. Instrum. 22 (1959) 4%-490. Printed in the UK
Hot-wire di rection-error
response equations in two-
dimensional flow
Fumio Yoshino, Ryoji Waka and Tatsuo Hayashi
Department of Mechanical Engineering, Tottori University, Minami 4-101, Koyama, Tottori 680, Japan
Received 14 October 1988, accepted for publication 31 January 1989
Abstract. In order to express explicitly the effect of the angle-errors of setting a hot-wire probe in a flow field on the output of the hot-wire anemometer, approximate hot- wire response series equations based on the small angle- error and weak turbulence approximations (called direction- error response equations in this paper) were derived from I-type, slanted-type and X-type hot-wire probes in two- dimensional laminar and turbulent flows from a general hot- wire response equation.
were confirmed experimentally as being able to predict accurately the effect of the angle-errors on the output of the hot-wire anemometer as long as these angle-errors were small. It was found from these experiments and the theory that the reasoning in treating the yaw factor as a function of the yaw angle is weak and not well founded.
The characteristics of the direction-error response equations for turbulent flow are discussed in terms of the order-of-magnitude theory and certain specific turbulent flows. It was found that both the magnitude and relative accuracy of the output errors due to the angle-errors predicted by the theory were dependent on the direction of the respective angle-errors and the Reynolds stresses of interest.
that some of the angle-errors appear in these equations in the form of the sums of the angle-errors of specific directions regardless of laminar or turbulent flow.
The direction-error response equations for laminar flow
It was found from the direction-error response equations
1. Introduction
Hot-wire anemometers still play a major role in measuring turbulent flow, but they are hard to use, and the data ob- tained usually show a large scatter. One of the reasons for this is that hot-wire probes are not only very fine but also that the response to the change of direction of a hot wire in a flow is sensitive (Gilmore 1967).
Many investigations have been carried out on directional sensitivities (Champagne et al 1967, Champagne and Sleicher 1967, J0rgensen 1971) and on aerodynamic interferences of prongs, stems, etc (Compte-Bellot et a1 1971, Strohl and Compte-Bellot 1973, Wygnanski and Chih-ming 1978), so that a hot-wire response to the directional change of a hot wire in a flow field can be almost precisely predicted. However, concerning investigations on the output error (referred to as direction-error hereafter) caused by the error of angles (referred to as angle-error hereafter) which take place when setting a hot-wire probe in a flow field (Strohl and Compte-Bellot 1973, Klatt 1969, Kita 1979, Mueller 1982),
applications of their results are limited to certain particular angle-errors, or sbmetimes the results themselves are in- accurate. The main reason for this is considered to come from the fact that each researcher uses his own direction angle system limited only to his own problem which results in the limited application of the results. Yoshino er a1 (1986), however, were able to derive a general hot-wire response equation to show the output of a hot wire set at an arbitrary direction in a flow field. This general response equation, though derived for a different purpose, enabled us to discuss generally the direction-error characteristics of a hot-wire anemometer.
However general the response equation is, many factors must be taken into account in a real application as follows: the type of probes (I-type, slanted-type (which is referred to as S-type, hereafter), X-type and so on), the way of setting the probe in a flow (end-flow or cross-flow setting), and conditions of the flow field (laminar flow, turbulent flow, turbulence intensity and so on). This limits the generality of the results according to the conditions applied, though it is desirable that the constraints are as loose as possible. In the above mentioned end-flow setting, the mean velocity vector is in the plane formed by the hot wire and the prongs (referred to as the prong-plane, hereafter). In the cross-flow setting, the vector is perpendicular to the prong-plane.
The flow field considered here is limited to a two- dimensional flow since this is better known and since the measured higher-order velocity correlations are hard to find in a three-dimensional flow. The mean velocity is assumed to be measured by an I or an S probe while the turbulence components are assumed to be measured by an X probe at an end-flow setting.
The response equation is expanded to and approximated by equations which have explicit angle-errors (referred to as direction-error response equations, hereafter) because the influence of angle-error on direction-error cannot otherwise be seen without numerically computing this at a particular angle-error
.
The direction-error response equations in laminar flow were experimentally confirmed, and those in turbulent flow were numerically treated in terms of isotropic turbulent flow with normal distribution and experimental results obtained with a two-dimensional free jet by Heskestad (1965) (here- after referred to as isotropic turbulence and Heskestad's jet, respectively).
2. Response equations
2.1. General response equation
It is assumed that a constant temperature hot-wire anemo- meter and a lineariser are used, so that the output voltage E is proportional to the effective cooling velocity U, as follows:
E = SU, (1)
where S is sensitivity.
U, is expressed as a function (Jorgensen 1971, Hinze 1975) of velocity components U,, perpendicular to the hot wire and in the prong-plane, U, tangent to the hot wire, and U , perpendicular to both U,, and U, as shown in figure 1, that is,
where k and h ( = l + A h ) are yaw and pitch factors, respectively.
The velocity U', = (U,,, U,, U,,) in the coordinates fixed to a hot wire and prongs is related to the velocity UT = ( U , V , W )
W
/’
XF
X
Figure 1. Coordinates, angles and velocity components. in the coordinates ( x , y , z ) fixed to a flow field as follows:
U , = QU (3)
where Q-has the following elements Q,: Qll = sin @ cosp cosy- sin @ sin a sinp sin y
+
cos @ cos a sin yQ12= sin @ cosp sin y + sin @ sin a sinp cosy
-
cos @ cos a cos yQ I 3 = -sin @ cos a sinp-cos @ sin a = cos @ cosp cos y - cos @ sin a sinp sin y
- sin @ cos a sin y
Q2’= cos @ cosp sin y
+
cos @J sin a sinp cosy+
sin @ cos a cos yQ23= -cos @ cos a sinp+sin @ sin a Q31 = sinp cos y+sin a cosp sin y Q32 = sinp sin y
-
sin a cosp cos yQ,, = cos a cos
p.
(4)The angles a,
p,
y and @ are shown in figure 1. a is the rotational angle around the probe stem,p
is the deviation angle of the prong-plane from the probe stem( p
is drawn in the negative direction), y is the rotational angle around a line parallel to the z axis, and @ is the angle of intersection of the direction of the prongs with the hot wire. When a= p
= y = 0, the prong-plane is on the x-y plane and the probe stem is in the direction of the x axis.Substituting equations (3) and (4) into ( 2 ) yields the relation of U, to the mean velocity components U ,
Y ,
W and the turbulence velocity components U , U , w in x , y and zcoordinates as follows: U U w a u ’ b u 2 U,=
TU
(
1 +2q-+2r-+
a-+--
+--
U U U T Z U 2 T Z U 2 c w’ d uu f uw g wu t’ U2 2 ’ 0 2 t2 U’ t2 U’ +-T+--+-7+-- where a = Q i l+
k’Q&+
h2QZ1 b = Q:2+
k’Q ;2+
h’Q :2 c = Q :3+
k‘Qi3+
h2Q& d=2Q11Q12+
2pQ21Q22+
2h2Q3iQ32f
= 2&12(213 +2k2Q22Q23+
2h2Q32Q33 g = 2Q11Q13+
2k2Q2iQz3+
2h2Q31Q33t 2 = a
+
b tan’e,
+
c tan’ 8 ,+
d tan 0,+
f tan 8” tan 8,+
g tan 8 ,q = (2a
+
d tan 8, + g tan f3,)12t2 r = (26 tan 8,+
d+
f tan 8 , ) 1 2 ~ ~s = (2c tan
e,+
f tane,
+g)12tztan 8,= VIU tan 0, = WI
U .
( 6 ) Substitution of equation ( 5 ) for U , in equation (1) pro- duces a hot-wire response equation. However, equation (5) together with (4) and (6) is called here the hot-wire response equation since E is proportional to U,. Equation (5) can give the output of the hot wire at an arbitrary angle in a flow field. The probe-setting in a flow field may be an end-flow type or a cross-flow type. Equation (5) is in that sense the general response equation (Yoshino et a1 1986).2.2. Direction-error response equations
Here equation ( 5 ) is also expanded into a series by assuming that the absolute value of the sum of the terms, except for the first term in parentheses, is less than unity because otherwise there seems to be no way to separate the turbu- lence velocities from the mean velocity.
The effective cooling velocity U,, the left-hand side of equation ( 5 ) , consists of U e , the mean value component, and U,, the fluctuating component. Equation (5) expanded into a series is separated into two series equations, one is that of a mean value component and the other is that of a fluctuating component. The terms with velocity correlations higher than second order are neglected in the series of Uc while those with velocity correlations higher than fourth order are neglected in the series (called fourth-order exact response equations, hereafter) of U: etc since measured turbulence velocity correlations higher than fourth order have not been found in the literature.
Angle-errors are represented by putting A in front of letters, such as ha, AB, Ay and A @ . The 8, deviation angle of the mean velocity from the x axis is here regarded as an error like angle-errors. Angle-errors, B u , Ah( = h
-
1) and the RMSvalues of turbulence velocities such as
(z)1’2/i3
are all assumed to be of the same order of magnitude as k , the yaw factor. k and A h of commercial probes are usually of the order of 0.1. In our own experience when a probe is carefully set by measuring the angles with the human eye, the angle-error can be assumed to be less than 5”(0.0873 rad). Values of BU of the order k( = 0.1) can cover most of the flow field even in the case of a two-dimensional jet (Heskestad 1965). Therefore, the assumption of the order of magnitude of errors, such asF Yoshino et a1
angle-errors, is not unreasonable. In the process of the derivations of the following relations, sin A a , tan 6 , and cos A a , for instance, are approximated by A a , Bo and 1 - t A a z , respectively. The subscripts 1 and 2 in the direction- error response equations derived below correspond to the hot-wire probes with $ = 45” and 135” respectively in the case of an S or X probe. After lengthy calculation, the following direction-error response equations were obtained.
2.2.1. Laminarflow equations. The response equation for an
I probe is as follows:
U a / U 2 z [ 1 -
k 2 - 2 ( A $ + A y ) 2 - A a 2 + 4 ( A $ + A y ) 8 , - 8 ~ ] x c o s 2 y a + [ 6 , - ( h $ + A y ) ] sin2y,+k2+(A$
+
Ay)’+ Aaz-2(A$+
Ay)@,+ 6: (7)while that for an S probe is expressed by
i T ~ l U 2 ~ + [ f 4 ( A $ + A y ) - ( A a T A p ) 2 + 2 A ~ Z T 4 8 , ] c o s Z ya
if [ 1
-
k2-
2( A$+
Ay)2 - +( A a T AP)’ +4(A$+Ay)8,-82,] sin2ya++[1 + # T 2 ( A $ + Ay)
+
( A a 7 AD)*-
AD2 i 26,+
et]
(8)where the upper and lower signs of the double signs corre- spond to the hot wires with $ = 45” and 135” respectively and
ya is the apparent value of y.
2.2.2. Turbulent flow equations. In the case of turbulent flow, two types of probe are considered, i.e. I and X probes. Equations to obtain the mean value U and the
x
component of the Reynolds stresses U * are derived for the I probe, while equations to obtain the Reynolds stresses are derived for the X probe. The apparent value ya is fixed to zero here since it is the most common case in turbulence measurement, and since k and h are not determined in turbulent flow.The response equations for U and u2 by an I probe are as follows, respectively: - W2 U (9) U e
_ -
U
-
1 -+(A$+
A Y ) ~ + (A$+
A y ) 6 , + 7 and --
-
U: U 2 U 2 uvF==
[l-
(A$+
AY)’]F~+
(A$+
A Y ) ~ T U2+
2(A$+
A y ) s-
-
- -
uw2 uw2 1 w4 U Z W 2 U3 U4u
u4
+
(1+
2 A h ) ~+
(A$+
Ay)--?+
- r4-
7 1 ( 2 ) 2 4U4
.
The following response equations are all for the X probe The response equations for U’ and
7
are as follows: (ue2*
Ue1)’U2
to obtain the Reynolds stresses_
=[(a)+
(A)(G+k?)-(i)(A$2+ A Y ~ - A $ ~ - A Y I )-
U 2
Ti(A$Z+AY2+A$l+ Arl)’+(b)(AP:+AP:)I~
where signs like f should be taken to refer to the respective upper and lower numerical values in parentheses in the same order as given and the upper and lower signs correspond to the equations for
2
and3,
respectively.The response equation for
2
is-
M U
+ A a z + A ~ 2 + A a , - A ~ l ) ~ + 4 ( A y 2 + A y 1 + A a 2
U
Finally, the equation for UV is as follows:
- -
2 2
U e z - Me1
[2 - (ki
+
k:)-
2(A@z+
Ay*)’- 2(A@1+ AyJ2-3
U2
-
U V
- $ ( h a 2 + AP2)’- +(Aal
-
A/~,)*]T + + [ k i - k?-
U 2-
2( A42+
Ay2+
A@ 1+
Ay1)+
AD-
AB :]r2
+
$ [ 3 (k:-
k:)+2(A$2+Ay2+A$l+Ay1) -(Aa2+AP2)2+(Aat-AP1)2 - AD:
+
A / 3 ? ] 3+
I [ ( A a 2+
APz)2-
( A a l-
AP1)’]- U2-
-
V 2 W 2 U W ? +2(Ah2-Ahl-8,),+2(1 U + A @ ~ + A Y ~ - A $ ~ - A ~ ~-
-uwz uuw2 + A h z + A h l ) ~ - 4 - Uu4
’It is apparent from all the above response equations that some of the angle-errors appear in combination such as A a l - ABl, A a z
+
AB2, AG1+
Ay1 and AGZ+
A y 2 with a few exceptions. This implies that even small angle-errors in ab- solute value could result in a large direction-error if they accumulate.w
Figure 2. Probes used in the experiment.
3. Experimental results and discussion
In the case of laminar flow, U and 8, are easy to measure and the number of combinations of angle-errors is relatively small for the I and S probes; hence the characteristics of the direction-error are experimentally determined. On the other hand, in the case of turbulent flow,
0,
e,,
the Reynolds stresses and the higher-order turbulence velocity correlations are all hard to measure accurately. Furthermore, the number of combinations of angle-errors is prohibitively large for an X probe. Therefore, in turbulent flow, the results calculated from the direction-error response equations are compared with those calculated from the fourth-order exact response equations on the basis of an isotropic turbulence and Heskestad's jet, since this comparison can still provide us with an approximate estimation of the accuracy of the direction-error response equations as will be shown later. 3.1, Direction-error in laminar flow3.1.1. Experimental apparatus and method. The experiment was made in the core region of a circular free jet. The exit diameter of the nozzle was 30 mm and its contraction ratio
was 91100. The Reynolds number based on the exit diameter
was 6.6 x lo4 _while t h e turbulence intensity was q21
U 2
-
0.0004 where q 2 = ) ( u 2 + v 2 + w'). With a long period of the mean velocity and temperature, fluctuations were less than
1% of the mean velocity and 1 K , respectively. The probes were made as described by Compte-Bellot et a1 (1971),
Strohl and Compte-Bellot (1973), Jerome et a1 (1971) and Wygnanski and Chih-ming (1978) (figure 2). The sensitive part of the hot wire was made of tungsten wire of 5,um diameter and 1 mm length, and both ends of the wire were copper-plated to about 50pm diameter. The anemometer and lineariser were DISA-55D05 and DISA-55D10, respectively. The A$ values were measured with a microscope. The a ,
p
and y values were determined using a telescope and a specially made jig, with accuracy of angle measurement better than 0.2". The zero point of ya was obtained from the point of the maximum value of the curve through output data measured within the range of ya = 90"-
(@+
A$) -t 10". This curve was determined by a least-squares method. The zero ofya when 8, # 0 is the direction of ya =
-
8,.The measurements were made at the Reynolds number
11, based on the wire diameter, and at an overheating ratio
of 0.8. Probes with a slack wire or drift were not used. Measurements were made on I, S and X probes with h a , hp and 8, as parameters. The results of the experiment for all types of probes showed good agreement with the theoretical values predicted from equations (7) and (8). However, the results of the S probe only = 45") are shown here. 3J.2. Direction-error characteristics. The ordinate t of the
figures below is the ratio of Ue to
U ,
that is, the square root of equation (8). The theoretical curves in the figures are the curves drawn by using equation (8).Figure 3 indicates the comparison of the theoretical pre- dictions with the experimental results when both ha and hp values exist for the S probe with @=45". In this case, equation (8) is rewritten as
(Aa
-
A@)' 1 - sin 2y,0%
l+sin2y, 0 2 - 1 2+(@+
2_ -
2 2 6 p 2-
( A a-
AB)* 4 cos 2ya. (14)+
- 60 -30 0 30 60ra
Slanted wire 1Figure 3. Comparison of theoretical predictions with experimental results for values of Aa and A@. R e = = l l , 8,=0, k = 0 . 0 8 3 , A + = 2 " ,
F Yoshino et a1
At ya =
-
45", the direction-error is zero when A/3 = A a , but is2 A a 2 when AB =
-
A a . However, equation (8) shows that the direction-error at ya=45" is 2 A a 2 when A p = A a for the Sprobe with
C#J
= 135".Direction-error in the case of 8, # 0 is shown in figure 4. In this case, equation (8) is rearranged to
I I I I I I
-60 -30 0 30 60 90
idegl Slanted wire 1
Figure 4. Comparison of theoretical predictions with experimental results for values of Bi;. Re = 11, k = 0.083, A@ = 2", A y = O", Au = 0", Ab = 0".
The theoretical predictions and experimental results agree well with each other. The theory shows that the direction- error is large when ya = 0" and that the measured value of k is directly affected by
e,.
A case of co-existence of A a , AB and 8, is shown in figure 5. Although the theoretical calculations in this case also agree well with the experimental results, the former deviate slightly from the latter at a small value of ya when 0, is large (symbol U). This discrepancy is considered to come from neglect of the terms with degrees higher than the second degree of k in equation (8) because the calculated result from the general response equation ( 5 ) agrees well with the experi- mental result in this case, as well.
I
0 ---zoo- 60 -30 0 30 60 90
Slanted wire 1 v0 1%)
Figure 5. Comparison of theoretical predictions with experimental results for values of h a , AD and 0,. R e - 11, k=0.083, A@=2', Ay=O", AU = -lo", A b = 10".
Table 1. Maximum angle-errors allowed for slanted wire.
2.6 0 0 0 0 0 2.4 0 0 0 0 0 0.4 0 0 0 0 0 0.4 0 0 0 0 0 0.8 6.6 6.6 0 0 0 1.4 -1.4 0 0 0 0 0 0 14.4 14.4 0 0 0 -0.4 0.4 6.0 6.0 6.0 6.0 6.0 1.4 -1.4 1.4 1.4 1.4
-
43.2 - 42.6-
39.4-
39.0-
38.8 - 39.4 -43.4 - 24.8-
39.2 - 33.4-
42.0 5.38 5.02 5.32 5.32 5.04 5.12 6.16 5.04 5.05 5.11 6.19It is obvious from the above results that when angle-errors and 0, are as small as k , equation (8) can express the direction-error well. The same degree of accuracy of the theory is confirmed for the I probe, the S probe with @ = 135"
and each wire of the Xprobe, though experimental results are not shown here.
Table 1 shows the maximum angle-errors allowed for the S probe with @ = 45" to cause a direction-error of less than about 5% and the direction ya when that direction-error occurs. It is seen from the table that A# and Ay should be less than 0.4" and that A a and A/3 should be less than 1.4" to keep the direction-error less than about 5%.
3.1.3. Dependence of the y a w factor on yG. The values of the yaw factor k used to draw the theoretical curves in the above figures are those measured when the hot wire is parallel to the streamline and they are assumed to be independent of ya. However, it is natural to consider the value of k as a func- tion of y2 (Jergensen 1971). Figure 6 indicates repeatedly measured values of k for the S probe. From this experiment, the values of k appear to be a function of ya, but it is also obvious that the data are more scattered when y2 becomes larger. The same tendency is seen in Jergensen's experiment
(1971). k values in figure 6 were determined from the follow- ing equation: 2 ( 0 3 U 2 )
-
(1 +sin 21,) 1-
sin 2ya ka = (16) 1qPn
0.4- k 0.2 - 4 5 -30 -15 0 1 5 30 45 Ided Slanted wire 1Figure 6. Measured values of the yaw factor ( S probe). R e = l l The bar represents twice the standard deviation.
where k, in equation (16) is the apparent k and
0i/U2
is a measured value. The effects of the angle-errors on k, for8, = 0, are obtained from equations (16) and (8): k:= k2+2(A$
+
A y ) 2 + i ( A a - A/3)2+ [2(A$+
Ay) -:(ha' (17) ( A $ + AY)* - A p ) 2 + A p 2 ] tan(ya+n/4)- ,
sin2(ya - n14) k, values with some angle-errors obtained from equation (17) for k = 0 . 1 are shown in figure 7. The figure shows that the values of k, deviate further from the true value of k when the angle of the hot wire approaches a right angle to the streamline (ya=45" in this case). Since k,>O, an experi- imenter tends to exclude the experimental data with k,<O
from his list of data. Also, it is probably impossible to keep the angle-errors zero. As a result of these factors, it is concluded that a measured value of k, has a tendency to be k a > k , and that this tendency becomes stronger when y a
becomes larger. Moreover, the same kind of equation as (17), though not shown here, can be derived in the same manner to show the effect of A E on k, where AE is the error which occurs when an experimenter reads the output E of a volt- meter. This equation also indicates the same tendency as figure 7. For instance, when AEIE is only 0.270, the error in k due to this exceeds k 100% for ya> 15". These facts do not deny the existence of a functional relation between k and y ,
but imply that the experimentally determined relation itself between k and y is uncertain and doubtful.
t
0.4,, I
t
x
I \ I I \ I I- 4 5 0 45
r n
l d e g lFigure 7. Values of the yaw factor influenced by the angle-errors. All of these factors plus the good agreement of the theoretical predictions with the experimental results as shown in figures 3-5 show that k may be regarded as a constant and that the reason for having to treat k as a function of y is practically weak and not well founded.
3.2. Direction-error in turbulent pow
3.2.1. Relative direction-error. In order to show more clearly the effect of angle-errors on the _Reynolds stresses, the relative direction-error equation for u2 is defined as follows: (18)
( u e 2
+
UeJ2(a)-
( U e z+
U e l ) * ( O )A1(?) =
-
2u*
where (ue2
+
uel)*(a) and (ue2+
U , ~ ) ~ ( O ) are the outputs first with then without the direction-error, respectively. The samekind of equations for relative direction-errors A l ( 7 ) , A l ( 7 ) and AI(%) can be defined, though they are not shown here.
As an experiment in turbulent flow was not carried out owing to the reason Alreadymentioned, the numerators of the equations for Al(u2), A l ( v 2 ) , etc were estimated from the fourth-order exact response equations. These are the original equations expanded into series having up to the fourth-order velocity correlation terms with no other approximations. The direction-error response equations (lo)-( 13) were derived from those series equations with assumptions of small angle- errors. Therefore, Al(u2) etc, obtained by using equations (10)-(13), were compared in the following with those values obtained by using the fourth-order exact response equations. The two types of turbulent flow used for error-estimation were isotropic turbulence with the intensity of q 2 / U 2 = 0.01 and Heskestad's jet as shown in table 2. Values in table 2 are those downstream by 100 times the nozzle width. 7 is a dimensionless distance from the symmetric plane of the two- dimensional free jet.
0
is approximately 87% of the mean velocity at the symgetric plane when 11 = 0.05, where the turbulence intensity q2/o2 is about 0.073.In the following figures, the values obtained from the fourth-order exact response equations (referred to as
f-
experimental values, hereafter) are shown with symbols
0,
A , etc and are compared with the theoretical curves ob- tained from the direction-error responseequationsfor a few combinations of angle-errors. Only Al(u2) and Al(v2) for an X probe are shown here owing to the limited space.
Figures 8(a) and ( b ) indicate A I ( u 2 ) caused by A a and Ap values in an isotropic turbulence and Heskestad's jet, respectively. In the figures, AD2 = AD,, for example, means that A/$ is also 10" when Apl is 10". The theoretical curves are drawn from the following relation obtained by substitut- ing equation (11) into equation (18):
-
-
-
u2-UU U Z + E
-
( Aal - Ap,)']+
-AD 2u2+
-Ab 2u2!.
Table 2. Heskestad's jet at 7 = 0.05.Velocity Velocity
correlations Values correlations Values
-
-
U w2l U3 - q213U2 0.0734-
21
-
0 2 0.1103-
U 4 1 U4-
U 2 1 U 2 0.0488-
U 4 1 U 4-
UUIU'Z 0.0266-
U3VlU4-
u3/ U3 0.0016 U4-
u 3 1 u 3 0.0014 u2w=/ 0 4-
u2u/U3 - 0.0004-
u2w2/ U4-
U U 2 1 U 3 0.0009Uu3/P
w4/ U4-
w21 U 2 0.0611-
UW21 U 3-
0.0003 uuw2IU~ - 0.0002 0.0341 0.0067 0.0105 0.0029 0.0054 0.0067 0.0003 0.0013 0.0016F Yoshino et a1
10 20 30
Angle-error ( d e g )
Theoretical curves agree well with f-experimental results except for the case of large angle-error in Heskestad’s jet.
Figures 9(n) and ( b ) show Al(u2) due to A@ values in an isotropic turbulence and Heskestad’s jet, respectively. It is obvious from equation (19) that relative direction-errors due to A$ values are much larger than those due to A a and AB values. In this case also, the discrepancy between the theoretical calculations and the f-experiment is a little larger in Hesketad’s jet than in an isotropic turbulence.
Figures lO(a) and ( b ) are Al(u2) due to A a and AB values in an isotropic turbulence andseskestad’s jet, respectively. Figures ll(a) and ( b ) show Al(uz) due to A@ values in similar flows. The theoretical curves of these figures are drawn from
( b )
I
! II
Angle-error i d e g )
Figure 8. Relative direction-errors of
7
due to Aa and Ap values. ( a ) Isotropic, ( b ) Heskestad. the following equation:LO 20
-
s
-
0-
-
a -20 -40 Angle-error I d e g lFigure 9. Relative direction-errors of
3
due to A@ values. (a) Isotropic, ( b ) Heskestad.-20 0 20 Angle-error [ de51
0 I
s
-
-10-
I>
a -2c 10) Angle-error ! degi i o 20 30Figure 10. Relative direction-errors of
7
due to Aa and AB values. ( a ) Isotropic, ( b ) Heskestad. which is obtained from equation (11) and the same kind 1 ,of equation as (18). Note that is eliminated since both ( ~ ~ ~ - u , ~ ) ’ ( a ) and (uez-uel)’(0) include the same 0”. In this case also, relative direction-errors due to A@ values are much larger than those due to Aa and AD values.
In most of the cases shown above, the approximate theory overestimates the relative direction-error, and the relative direction-error in Heskestad’s jet is larger than that in an isotropic turbulence.
3.2.2. Estimation of the accuracy of the direction-error res- ponse equations. As seen in the above sample calculations, the discrepancies between the theory and the f-experiment of
( b i
Angle-error i d e g )
the relative direction-errors are larger in
7
than in2
and larger in Heskestad’s jet than in an isotropic turbulence. As mentioned already, the fourth-order ‘exact’ response equa- tions themselves were equations which ignored the velocity correlation terms higher than fourth order, and hence include some truncation errors as an a priori fact. Therefore, the direction-errors obtained from the direction-error response equations do include errors due to the truncation of the higher-order velocity correlation terms as well as those due to other approximations such as small angle-errors. The order of magnitude of these truncation errors can, however, be esti- mated approximately by assuming an isotropic turbulence. As can be seen in table 2 , the third-order velocity correlations40 20
-
s
I 1%..
a -20 - 40 -2C 0 2 0 A n g 12 - e r - o r ; deg 1Figure 11. Relative direction-errors of
7
due to A@ values. ( a ) Isotropic, ( b ) Heskestad.-20 0 20
F Yoshino et a1
10
0.1
-
0.01 0.1
q T / O 2
Figure 12. Approximate residuals of the expanded response equations.
are relatively small in comparison with the second-order or fourth-order-grrelatjonsid the relations of isotropic turbu- lence such as u4 = 3 ( ~ ’ ) ~ , u 2 v 2 = u2 * vz, etc also approximately
hold for Heskestad’s jet. This suggests that Heskestad’s jet can be roughly replaced by an isotropic turbulence with an intensity equal to that of Heskestad’s jet. According to Taylor’s theorem, the order of magnitude of the truncated higher-order velocity correlation terms can be estimated by that of the lowest-order velocity correlation terms among the truncated terms. Hence, the residual of the fourth-order exact response equation may be estimated from the propor- tion of the fourth-order velocity correlation terms in the fourth-order exact response equation. Figure 12 shows those residuals in the case ofan isotropic turbulence as a function of turbulence intensity q 2 / U 2 . It is interesting to see that the residual of (ue2 - %)*( = 2 7 or 2 3 ) is much larger than that of (ue2
+
u , ~ ) ~ ( = 2u2). The former is 2.3%(19%) while the latter is 0.07%(0.47%) when q 2 / U 2 is O.Ol(0.07). This implies that both the contributions of the ignored higher-order terms to the direction-error calculated by the fourth-order exact response equation a n c b y the diLection-error response equa- tion are far smaller in U * than in U * . This is the reason_why the agreement between the relative direction-errors of u2 by the f-experiment and by the theory is better than that of7
as seen from the comparison of figures 8 and 9 with figures 10 and 11. T h e Sam2 comments a s b o v e can be applied to the case of w 2 and U*. Since U ~ ~ - U ~ ~ = O up to the fourth-ordervelocity correlation terms in an isotropic turbulence, the same kind of reasoning cannot be applied to the case of E. Figure 12 also suggests the reason for the discrepancybetween the theory and thef-experiment in Heskestad’s jet
($/U2=
0.07) to be larger than that in an isotropic turbulence(?/U2
= 0.01) -in figures 8-11 since the residuals become larger when
q 2 / U 2 becomes larger in all cases.
In all cases shown in the figures, the relative direction- errors due to Aq5 and Ay values are much larger than those due Act and AD values. According to equations (11) for u2
and v 2 and (13) for E, the lowest degrees of the terms with Aq5 and Ay values, A a and AD values and Ov in these equations are third, fourth and fourth, respectively when the degree is counted as third in terms such as Aq5u2. Since the contributions of the angle-errors are larger when the degree of the terms with these angle-errors is lower, the contribu- tions of the terms with Aq5 and Ay values, A a and Ai3 values
and 8, are largest, medium and smallest, respectively. This is the reason why the relative direction-errors due to Aq5 and Ay values (figures 9(a), (6) and l l ( a ) , (6)) are much larger than those due to A a and AD values (figures 8(a), (6) and 10(a), (6)). (The relative direction-error due to Ou is smallest but does not appear in the figures because of isotropic turbulence assumption.) The discrepancy between the contributions of the terms with A a and AD values and with BV occurs because the third-order velocity correlations are usually much smaller than the second-order correlations and because the terms with A a and A/3 values appear first in the second-order velocity correlation terms while those with OC appear first in the third-order velocity correlation terms.
The accuracy of the relative direction-error due to re- spective angle-errors can be estimated as follows: since the terms with the degree of k higher than fourth are all ignored in the direction-error response equations, the lowest degree of k ignored in the fourth-order velocity correlation terms (which do not include the angle-errors as seen from equations (11)-(13)) is fifth. This means that the degree (fourth) of the terms with ha and AD values is only one order lower than that (fifth) of the terms ignored while the degree (third) of the terms with Aq5 and Ay values is two orders lower than that (fifth) of the terms ignored. This results in the contributions of the fourth-order velocity correlation terms to the relative direction-error being larger in the case of A a and AD values than in the case of A$ and Ay values when they are estimated from the fourth-order exact response equation. Figure 13 shows the proportions of the contributions of the fourth-order velocity correlation terms to the relative_direction-errors due to respective angle-errors obtained for v2.
As can be seen from the above discussion, the contri- bution is smallest in Aq5, medium in A a and AD, and largest in
O u , which means that the relative direction-errors predicted by the direction-error response equations are most accurate in the case of Aq5 and A*/ values, less accurate in the case of A a and AD values and least accurate in the case of
O D
within the range of assumptions made.
For w 2 , the situation is different from that for
2
and7
since equation (12) shows that the lowest degree of the terms with 8, is equal to that of the terms with A a and AD values. Consequently, the contribution from the fourth-order velo- city correlation terms to the whole relative direction-error in the case of 8” is about equal to that in the case of ha and AD100
r
I t1
-? / U -?
0.01 0.1
Figure 13. Contribution of the fourth-order velocity correlation terms to the whole relative direction-errors of U*.
10 20 30 Angle-error I d e g )
-2 0 0 20
Angle -error ldeg)
Figure 14. Relative direction-errors of U0 in Heskestad's jet. ( a ) Those due to Aa and A@ values. ( b ) Those due to A@ values.
s l u e s . This was confirmed by the same kind of curves for w2 as found in figure 13, though these are not shown here.
The above discussion- is sum_marised as follows: the rela- tive direction-errors in U' and v 2 predicted by the theory are (i) largest but most accurate in the case of A@ and Ay values, (ii) medium but less accurate in the case of ha and AD values and (iii) smallest but least accurate in the case of 8,. The relative direction-errors in w 2 are different from the above only in that those due to 8, are categorised into (ii) of the above. Though the same kind of reasoning is not possible in the case of UU because of the reason stated already, figures 14(a) and ( b ) for the relative direction-error in Heskestad's jet suggest that the accuracy of the theory for UV is about equal in order to that for
7.
It should be noted finally that Heskestad (1965) does not mention the accuracy of setting the hot-wire probes and that some of the turbulence velocity correlations of Heskestad's jet were not measured but inferred theoretically.4. Concluding remarks
A general hot-wire response equation (5) was obtained for a constant temperature anemometer with a lineariser.
From this response equation, the direction-error response equations (7) and (8) were derived for a two-dimensional laminar flow. From the comparison of the theoretical predic- tions with the experimental results, the following conclusions were obtained:
(i) equations (7) and (8) were confirmed to be very accurate as long as the angle-errors are small,
(ii) in order to keep the direction-error less than about
5 % for the S probe with @=45", it is necessary to keep Ay
and A@ less than 0.4", and ha and AD less than 1.4", respectively,
(iii) the yaw factor k may be regarded as a constant and the reason for having to treat k as a function of y is weak and not well founded.
From the general hot-wire response equation (5), the direction-error response equations (9)-( 13) were derived for
a two-dimensional turbulent flow. From the comparison of the predictions by these equations with the f-experimental results and from the order-of-magnitude theory, the charac- teristics of the direction-error response equations can be summarised as follows:
(i) the direction-error respogeLquation for U * is more accurate than the equations for U ' , w 2 and E.
-
(ii) the relative direction-errors in u2 and U * are largest but most accurate in the case of A@ and Ay values, medium but less accurate in the case of ha and AD values and smallest but least accurate in the case of 8,. T& relative direction- errors in w 2 are different from those in U' and v 2 only in that the relative direction-errors due to 8, are approximately equal in magnitude and accuracy to those due to Aa and AD values.
(iii) In both laminar and turbulent flows, some of the angle-errors appear in combination such as hal - AD,, Aa2
+
AD2, A@l+
Ay,, Ay2 with only a few exceptions. These combined angle-errors could result in a large direction-error if they accumulate.-
References
Champagne F H and Sleicher C A 1967 Turbulence measurements with inclined hot-wires. Part 2 . Hot-wire response equations
J. Fluid Mech. 28 177-82
Champagne F H , Sleicher C A and Wehrmann 0 H 1967 Turbulence measurements with inclined hot-wires. Part 1. Heat transfer experiments with inclined hot-wire
J . Fluid Mech. 28 153-75
Compte-Bellot G, Strohl A and Alcaraz E 1971 On aerodynamic disturbances caused by single hot-wire probes Trans. A S M E : J. Appl. Mech. 38 767-71
Gilmore D C 1967 The probe interference effect of hot-wire anemometers
F Yoshino et a1 J. Phys. E: Sci. Instrum. 22 (1989) 49C-498. Printed in the UK Heskestad G 1965 Hot-wire measurements in a plane
turbulent jet
Trans. ASME: J. Appl. Mech. 32 721-34 Hinze J 0 1975 Turbulence 2nd edn (New York: McGraw-Hill) p 127
Jerome F E, Guitton D E and Pate1 R P 1971 Experimental study of the thermal wake interference between closely spaced wires of an X-type hot-wire probe
Aeronaut. Q. 22 119-26
JQrgensen F E 1971 Directional sensitivity of wire and fiber- film probes
DISA Information 11 31-7
Kita Y, Hikage N and Hirose K 1979 A study on a measuring method of flow by an X-wire anemometer Trans. JSME 45 437-40 (in Japanese)
Klatt F 1969 The X hot-wire probe in a plane flow field DISA Information 8 3-12
Mueller U R 1982 On the accuracy of turbulence measurements with inclined hot-wire
J . Fluid Mech. 119 155-72
Strohl A and Compte-Bellot G 1973 Aerodynamic effects due to configuration of X-wire anemometers
Trans. ASME: J . Appl. Mech. 40 661-6
Wygnanski I and Chih-ming Ho 1978 Note on the prong configuration of an X-array hot-wire probe
Rev. Sci. Instrum. 49 865-6
Yoshino F, Waka R and Yasui M 1986 The output error due to the errors of angle of hot-wire probes
Trans. JSME 52 3498-503 (in Japanese)
The behaviour
of a thermal-
gradient sensor in laminar and
turbulent shear
flow
B W van Oudheusden
Laboratory of Electronic Instrumentation, Delft University of Technology, Department of Electrical Engineering, PO Box 5031, 2600 GA Delft, The Netherlands Received 22 December 1988, accepted for publication 17 February 1989
Abstract. This paper describes a thermal sensor which detects flow on the basis of a flow-induced temperature gradient on its surface. A theoretical model has been developed to investigate the influence of design parameters such as sensor dimensions and material properties on the sensitivity and the time response of the sensor. The behaviour of the sensor has been analysed for laminar and turbulent shear flow, showing that the presence of
turbulence will increase the total heat transfer, while reducing the induced temperature difference. Flow experiments have been performed in laminar and turbulent flow with an integrated silicon sensor, and the results of the tests were found to be in good agreement with the
predictions from theory.
1. Introduction
The aim of the present study is the theoretical modelling and the testing of a thermal flow sensor, which can be made in the planar silicon technology used for the fabrication of inte- grated electronic circuits (Huijsing et a1 1982, Van Oudheusden 1988). The use of this technology allows the realisation of complicated sensor structures, while in addition the sensor may be combined with some signal-conditioning electronics, e.g. for signal conversion to obtain a sensor with digital or frequency output. The operation of the sensor under consideration relies here on the differential cooling of the heated sensor by a flow over its surface. The inhomoge- neous cooling effect of the flow induces a thermal gradient in the sensor in the direction of the flow. Consequently, flow can be measured by detecting a temperature difference on the sensor surface. In the present sensor, this operation is per- formed by an integrated thermopile. When compared with the traditional thermal anemometer operation based on the total heat loss of the sensor, the measurement of a tempera- ture difference has the advantage that it gives a zero output signal in the absence of flow and that the signal changes in polarity when the flow direction is reversed. Therefore, the method has a well defined zero-point, while in addition it can be used for directional flow measurement with a full 360" direction sensitivity (Van Oudheusden and Huijsing 1989),
To obtain a better understanding of the operation of the sensor, a theoretical and experimental study of its behaviour in shear flow was undertaken. Some preliminary results have been reported in an earlier publication (Van Oudheusden and Huijsing 1988). As the operation principle is closely related to the use of hot-film sensors for the measurement of shear stress, the theoretical description of these devices will be 490
