Errors in measurement of flow by velocity area methods

H. Botma Delft Hydraulics Laboratory, Delft, The Netherlands and A. J. Struyk Rijkswaterstaat, Dir. Upper Rivers, Section River Studies, Arnhem, The Netherlands

Abstract. Routine measurements of flow in open channels, carried out using current meters and applying the velocity area method, do not give sufficient information to estimate the accuracy of the method. In order to determine the magnitude of the errors much more detailed measure- ments are needed. This paper deals with the analysis of these detailed measurements.

Attention has been paid to errors due to: 1. Using a finite time to measure the local point velocities; 2. Using a finite number of points per vertical; 3. Using a finite number of verticals. Statistical methods have been used extensively in the analysis. Results of the analysis of

measurements in the River Yssel will be presented. This work is part of a project for the International Organization for Standardization.

ERREURS DANS LA MESURE DES DEBITS PAR LA METHODE DES ISOCHROMES

Résumé. Les mesures courantes d'un écoulement dans des canaux découverts, effectuées B l'aide d'un moulinet par la méthode de l'intégration du champ des vitesses, ne donnent pas d'informa- tions suffisantes pour estimer la précision de la méthode. Pour déterminer la grandeur des erreurs on a besoin de mesures plus détaillées. L'article décrit la méthode d'analyse de ces mesures plus détaillées.

On étudie avec soin l'analyse des erreurs dues B : (a) l'emploi d'un temps limité pour la mesure de vitesses en un point ; (b) l'emploi d'un nombre limité de points par verticale ; (c) l'emploi d'un nombre limité de verticales.

Un usage considérable des méthodes statistiques a été fait dans les analyses. On donne les . résultats des analyses de mesures dans le fleuve YsseL Ce travail fait partie d'un projet de l'organisation internationale de normalisation.

ERRORES EN LA MEDICI~N DE CAUDAL POR MÉTODOS DE ÁREA VELOCIDAD Resumen. Las mediciones de rutina de caudal en cauces abiertos, realizadas utilizando medidores de corriente y aplicando el método de área velocidad, no dan información suficiente para estimar la precisión del método. A fin de determinar la magnitud de los errores se precisan mediciones mucho más detalladas. La comunicación trata del análisis de estas mediciones y presta particular atención a los errores debidos a: 1. Uso de un tiempo finito para medir velocidades puntuales locales. 2. Uso de un número finito de puntos por vertical, y 3. Uso de un número finito de verticales.

En el análisis se han utilizado profusamente métodos estadísticos y se presentarán los resultados del análisis de mediciones en el río YsseL Este trabajo es parte de un proyecto para la Organización Internacional de Normalización.

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77 1

H. Botma and A. J. Struyk

NOTATIONS

h

'i

di E h(x) m

P 4 Q

S t

Qi

t0

'i

width of channel width of ith vertical depth of ith vertical ensemble average or mathematical expectation dapth profile number of verticals in a cross-section number of points in a vertioal unknown discharge calculated discharge calculated discharge of ith segment sampling error time duration of averaging instantaneous velocity measured velocity

v(x , y) velocity field over width and depth

'ik

V i

V( t)

vt:

-

'ik -

X

X Y

E( 1 h

"* "f 5 P* u* U' i 7

true velocity in point k in vertical i

true mean velocity in vertical i instantaneous local point velocity measured velocity in point k in vertical i

calculated mean velocity in vertical i distance from one of the banks measured value distance from water surface weight of velocity in point k relative root mean squared error of error j inverse of eulerian integral time scale mean of * relative mean of error j true value autocorrelation function of * standard deviation of * relative standard deviation of error j time displacement.

(Variables with and without errors are denoted by upper case and lower case symbolsres- pectively.)

772

Errors in measurement of flow by velocity area methods

INTRODUCTION

The unknown discharge 4 is the integral of a velocity field over a cross-section (Fig. 1).

4 = ~ ~ h ( J L ) V ( ~ , y) dx dy.

The use of the velocity area method means that the integral is approximated by finite summations. The mean velocity in a vertical i is approximated by the sum of p local point velocities K,'k, each with a weight ak.

P vi = r( ak . Vi&. k=l

The discharge 4 is approximated by a-sum of partial flows, products of the width bi, depthsdi and mean velocities Vi.

m ...

Q = bidi vi i= 1

(3)

Only errors which are characteristic of the velocity area method will be investigated. These errors are due to: 1. Sampling local point velocities in time, i.e. using a finite measuring time (error 1). 2. Sampling the vertical velocity profile, i.e. using a finite number of points per

3. Sampling the horizontal velocity profile and the depth profie, i.e. using a finite

Quantities relating to the errors will have a corresponding subscript. The errors are- supposed to be random, so in general the measured value X is the sum of the true value $ and a random sampling error S.

vertical (error 2).

number of verticals (error 3).

x=t+s. (4) S can be described by its relative mean y' -the -systematic part of the

error- and its relative standard deviation u', which are defined as the following over-all averages:

y' = E. {S}/x (5)

A yard-stick for both the systematic and the variable parts of the error is the relative root mean squared error E'.

773

H. Botma and A. J. Struyk

Figure 1 - Discharge as an integral of a velocity field over a cross-section

1.0

0.8

0.6

O. 4

0.2

0.0 I

2 4 6 10 - Ato Figure 2 - Graph for deviation u

The analysis of the errors is carried out by the Netherlands Technical Committee for the International Standardization Organization. Results are applied for measurements in the River Yssel in the Netherlands.

A somewhat similar investigation in this field has been carried out by Carter and Anderson (1963).

774

Errors in measurement of flow by velocity area methods

SAMPLING LOCAL POINT VELOCITIES (ERROR 1)

Model

Even, for steady flow, a local point velocity will fluctuate as the result of tur- bulence. Hence a measurement with a current meter over a short period yields a velocity which differs from the true mean velocity. This error has been called the fluctuation error. It is assumed that it has no systematic part, so pi = O.

The instantaneous local point velocity V(t) can be regarded as a stochastic process with time t as index. It will be assumed that this process is gaussian, i.e. for each finite number of times ti the corresponding random variables V(t,) have a multi-dimensional normal distribution, stationary, i.e. parameters are not a function of time, and finally ergodic, i.e. over-all averages are equal to time averages. Hence the mean pv, variance u; and the autocorrelation function pv(7) describe the process completely. They are defined as the following over-all averages.

Ily = E{V(t)) (8)

cJ$ = E{[ V(t) - pv]2) (9)

(10) Py(4 = E{[ Vit) - Pvl [Vt + 7) - Il,I)b$. Measuring with a current meter means the averaging of V(t) over a certain time to. So consecutively measured velocities Ui are linked with V(t) by

vi = - . siato V(t)* dt; i = 1, 2, 3,. . . (1 1) to (i-l)-to

It can be derived that the parameters of the time series q, (mean, variance and autocorrelation function) are the following functions of the parameters of V(t):

I-+ = Ilv (12)

These formulae can only be applied if pd~) is known. A useful approxi- mation is an exponential function, e.g. Hinze (1957).

py(7) = exp (-h . I T I) (15) in which h represents the inverse of the eulerian integral time scale. Substitution of equation 15 into equations 13 and 14 yields,(see Figs. 2 and 3)

775

H Botma and A. J. Struyk

0.6

0.4

0.2

o. o

\

-k Figure 3 - Function p(k)

o. o

o .2

0.4

z 10.6 5

2 0.8 ru

T ¡i l.ol 4

Figure 4- Some measured velocity profiles

776

Errors in measurement of flow by velocity area methods

From a measurement of a series of velocities Ui, with a certain to, it is possible to estimate the standard deviation and the autocorrelation function, Now there are two possibilities: 1. The estimated autocorrelation function agrees with the theoretical formula,

equation 17. Then h can be calculated with equation 17 and uv with equation 16. Hence uv can be determined for an arbitrary t, However, to should not be taken as too short an interval for the autocorrelation function (equation 15) does not hold when T is small compared with l/h.

2. The estimated autocorrelation function does not agree with the theoretical formula (equation 17). Then uv can be determined only for times longer than t,. For a duration of measurement of n - to, it can be shown that the standard deviation becomes

Measurements

Local point velocities have been measured uninterruptedly during 3,000 sec with the current meter being read every 30 sec. This has been done at 10 points in 3 verticals in a cross-section (verticals with 1; 0.6 and 0.3 times the maximum depth). The whole procedure was repeated for four different discharges. After- wards,a further measurement was added: 2,000 sec long, readings every 10 sec, in the above-mentioned verticals, but at fewer points. The result of a measure- ment can be regarded as a realisation of the time series Vi.

Analysis

After removing considerable linear trend in the series, due to unsteady flow, the relative standard deviation u1 and the autocorrelation function pv(k) were estimated. The normal distribution of the velocities was investigated with the x2 test, after a correction for the correlation between the velocities had been made.

Results

Table 1 gives some parameters of the cross-section depicted in Figure 5. Average results of the measurements with to = 30 sec and to = 10 sec are summarized respectively in Tables 2 and 3. Neither result is contrary to the exponential autocorrelation function (equation 15). Predicting u; for 30 sec from the results for 10 sec and vice versa gives good agreement with the model. Out of 143 ana- lysed series, the hypothesis that the measured velocities are normally distributed had to be rejected eight times (risk of 1 per cent).

777

H. Botm and A. J. Struyk

SAMPLING VERTICAL VELOCITY PROFILE (ERROR 2)

Model

The true mean velocity in a vertical F is approximated by vy the sum of p local point velocities V k with weights elk.

This can be regarded as sampling of the vertical velocity profile, resulting in a sampling error S,. With respect to an arbitrary velocity profile, this error has a random character. V is also influenced by the fluctuations of the measured local point velocities, resulting in a fluctuation error S, .

F= v+s, +s,. (20) Both S, and S, are dependent on the choice of the local points; SI is

also dependent on the duration of measurement. To separate the influence of the independent errors S, and S, repeated measurements of different velocity profïies are needed.

Measurements

Local point velocities have been measured consecutively for 60 sec at 10 points (near the surface and at relative depths of 0.2; 0.3; 0.5; 0.6; 0.7; 0.8; 0.9 nearer the bottom) in the deepest vertical of a cross-section. This was done 10 times without interruption. The total procedure was repeated for four different discharges. Mean velocities have been obtained with a planimeter. Some measured velocity profïdes are shown in Figure 4.

The results of these measurements can be summarized as Viik; i denotes different discharges, j repetitions at the same discharge, k local points in the vertical (i = 1, 2, . . . y I; j = 1, 2, . . . , J).

Analysis

It is supposed that the error in the mean velocity obtained with the planimeter can be disregarded. Measured velocities have been standardized by dividing by the corresponding planimeter velocity. Then all true mean velocities are unity and deviations from it are relative. The sampling error S, was assumed to have been a constant during the ten repeated measurements.

Then

Vj,j = 1 + s, I , i j + s2,i . The mean of the sampling error S, can be estimated by

778

Errors in measurement of flow by velocity area methods

Relative depth

u; x P"(1)

P"(2)

Table 1. Some parameters of the cross-section depicted in Figure 5

0.2 0.6 0.8 0.9 mean

6.2 9.6 11.2 13.6 10. I

0.27 0.30 0.33 0.25 0.28

0.00 0.12 0.12 0.04 0.07

Measurement

Discharge m3 is Area cross-section m2

Mean velocity mis

Maximal depth m

Width m

Mean depth m

21; 1 1;; 1 700

0.67 0.82 0.93

110

2.8 4.7 6.4

5.8 7.7 9.5

Table 2. Average results of measurements, to = 30 sec

1Rela;ive depth I '':J;'' ,I 0.2 10.3

0 1 x 3.9 4.5 I I ~ ~ ( 1 ) I 0.09 I 0.061 0.04

Table 3. Average results of measurements, to = 10 sec

779

H Botma and A. J. Struyk

The relative standard deviation of S,, however, is only a part of the standard deviation of q,i, the other part is due to the fluctuation error S,. It can be estimated by

I 1 1 2 . (vi,i - vi), (23)

1 I I - u, - - r( (Vi - 1 - p,) -

I- 1 I*J(J - 1) i=l

in which

Results

Several rules to calculate the mean velocities have been examined (Table 4). The standard deviation due to fluctuation and sampling error, a;+Z, is added to illustrate the effect of the separation, It is remarkable that the two-points rule is the best and that most means are significantly different from zero.

SAMPLING HORIZONTAL VELOCITY PROFILE AND DEPTH PROFILE (ERROR 3)

Model

The true discharge q is approximated by the sum of partial flow Qi, derived from depths, mean velocities and positions of a finite number of verticals. This implies a sampling error 3,.

n

i = l Q = Qi=q+S,.

This error is dependent not only on the number of verticals and their posi- tions, but also on the regularity of horizontal velocity profile and depth profile.

Measurements

A cross-section has been covered with 45 verticals at equal distances. In each vertical the depth and the velocity were measured at a relative depth of 0.6 for 120 sec. A reference velocity at one fixed point was measured to correct for unsteady flow. The procedure was carried out for four different discharges.

Analysis The error of the discharge determined by means of all verticals was supposed to be zero. Then the discharge was calculated using fewer verticals, and the relative deviations of the first discharge were analysed. The question arose as to which

780

Errors in measurement of flow by velocity area methods

Table 4. Examination of several rules for calculating mean velocities

Rules

- 1

2

3

4

5

6

7

8

9 -

.b

'' significant P O

- 4 x 3.6''

2.9

-

0.2*

1.3'.

1.9*

-1 .?

-0.9* *

-3.6

-0.8 .L

- 0:

7:

1.2

3.0

0.0

0.0

0.3

2.0

0.0

0.5

0.0

-

- E: x 3.8

4.1

0.2

1.3

1.9

2.6

0.9

3.7

0.8

-

- a:+* x 6.5

6.5

4.7

2.6

3.0

3.0

2.3

2.0

1.8

-

Table 5. Results of sampling horizontal velocity profile and depth profile

- m

- 4

7

12

23

-

- E: z 14.7

5.9

I .8

2.8

-

-

-8.8 6.1

-4.0 3.0

18 -0.6 1.2

27 0.0 1.3

- E; x 10.7

5.0

1.4

I .3

- i_ 1.8

781

H. Bofma and A. J. Struyk

verticals should be removed, and three ways of reducing the number of verticals were adopted: A. Keeping the distances between the verticals equal; B. According to the judgement of an experienced hydrographer who chooses his

verticals taking the depth profile and the distances between the verticals into consideration; so the depth profile has to be known in advance;

C. Keeping the partial flows as nearly equal as possible. This can be done, however, only when some previous knowledge of horizontal velocity and depth profiles is available.

Discharges were computed by the mid-section method, i.e. the product of depth and velocity is assumed to vary linearly with the distance between consecutive verticals. Results Results are summarized in Table 5 for methods A, B and C. Examples of thinning out patterms, according to methods B and C, are shown in Figure 5. A surprising fact is that the mean errors pi is always negative. The three methods

of choosing verticals do not give very different results. But for very few verticals only, it can be seen that the more information used in choosing the verticals, the better the result.

COMBINATION OF ERRORS

Supposing the three errors have a mean zero and are independent, it can be derived that they result in an error of discharge with a standard deviation ut

Use of this formula, with bzdi?i/q = l/m, taking values of e’ instead of u’ to be on the safe side, and using e; according to method B, has led to Figure 6. The number of verticals appears to have the most influence, which is not surprising if the form of the cross-section is taken into account.

CONCLUSIONS

Regarding Error 1 the following conclusions can be drawn: u; increases with incre- asing depth; no influence of the position of the vertical, discharge or other flow parameters on u; has been found; the theoretical autocorrelation function exp[- h . 1711 has not been rejected; and velocities measured with a current meter during 30, and even 10, can be considered to be normally distributed.

Regarding Errors 2 and 3 it should be realized that the results are based on measurements in one cross-section at four different discharges. To draw general conclusions more measurements are needed, and these have become available as a result of the IS0 investigations.

BIBLIOGRAPHY / BIBLIOGRAPHIE 1. Carter, R. W.; Anderson, I. E. 1963. Accuracy of current meter measurements. Proceedings

2. Hinze, J. O. 1951. Turbulence. New-York, McGraw-HilL 1957. A.S.C.E. Hy, VOL 4, July.

782

Errors in measurement of flow by velocity area methods

- E & 0.2

$ 0.4 G

.- c- O

2..

1:: 1.0

1.2

Y 0 - 3 0 E e

E & 0.2 c

- g 2

6

2..

0.8 8

1.0 - 10

1.2 - 12

0 - 3 0

- g 2 E

- 4

- 6 t

- 6

- 10

- 12

METHOD B

METHOD C Figure 5 -Patterns of velocity, depth and distance from the banks

at% 12

t 8

6

4

2

n

o í-PT 30 s

u 2-PT 30%

- 0 10 20 30 40 - NUMBER OF VERTICALS Figure 6 - Decrease of deviation with the number of verticals

783

H. Botma and A. J. Struyk

DISCUSSION

Lambie: Pulsating or fluctuating velocities may be caused by secondary currents due to the hydraulic conditions. Did you try to correlate the physical characteristics of the site with the hydraulic conditions in the approached channel and their effect on the pulsation itself?

Botma: We tried to correlate our results with the primacies of the cross-section, but we did not find any significant results. In this case we only used one cross-section under different circumstances.

Lamhie: A second question: When sampling local point velocities at ten points in each vertical, were the samples taken by ten meters simultaneously, or one at a time by the same meter? If simultaneous-readings were not taken, was an attempt made to take simultaneous reading at the more commonly used points, such as 0.2, 0.6 or 0.8?

Botma: In the investigations of the first error, we did measure simultaneously at different points in the vertical.

Lambie: A third question: At each site, has it yet been possible to detect the length of any short-term or long-term cycles in the pulsation period? And if so, has any conclusion been drawn from this on the optimum observation time?

Botma: We did not find any harmonic pulsation.

Hinrich: According to the rules, for the measurement of discharges in the Federal Republic of Germany each measurement shall last 30 sec. Do you think that 10 sec would be sufficient according to your observations ?

Botma: When using only two points, the exposure time should not be too short. If you are taking more points in a vertical, then the number of points averages the pulsation and you get a better result. If you are going from 30 to 10 sec then I think that it is not enough for the two-point method. With other methods at more points in a vertical, it may be enough.

Hinrich: W e do apply three different methods with the current meter: measurement at many points; measurement at two points; integrating measurement with a velocity of decline of 2 cmlsec. The accuracy is generally about * 4 per cent and in many cases even * 1 to 2 per cent. Botma: There are perhaps better methods to measure the velocity in a vertical. But you should realize that we had already taken the data, or the data were given to us by the ISO. Perhaps it would be very interesting to compare all these methods, but that was not the goal of the IS0 investigations.