Measuring directional velocity in water waves with acoustic flowmeter

MEASURING

DIRECTIONAL VELOCITY

IN WATER WAVES WITH

AN ACOUSTIC FLOWMETER

by

R. H. Muher

TECHNICAL MEMORANDUM NO. 31

APRIL 1970

U. S. ARMY, CORPS OF ENGINEERS

COASTAL ENGINEERING

RESEARCH CENTER

This document has been approved for public release and sale;

its distribution is unlimited .

Bibliotheek van de

Afdeling Schembouvi- en _Scheepvaartkunde

,Ter.ATCsche tiogeschool, Delft

DOCUMENTATiE

: K90-4

7

ABSTRACT

This report contains the technical details of an investigation which was undertaken to adapt an acoustical flowmeter to a device for measuring velocities in water-wave phenomena. The flowmeter studied was designed to measure the difference in travel times of two acoustical pulses travel-ing simultaneously in opposite directions along a common path. Because of viscous effects, a zone of low velocity flow occurs behind each probe and the measured velocity is somewhat less than the actual velocity when the

angle between the acoustical path and velocity vector, 6, is small. When this angle is relatively large the wake has little effect on the velocity and

Vmeasured = Vactual x cos°

It

is shown in this report that the wake effect may be eliminated by making simultaneous measurements along multiple paths.

FOREWORD

This report describes the effort to adapt the Westinghouse L. E. Flowmeter to laboratory and field investigation of water-wave phenomena. Some of the material in the report (Section II) was supplied through the courtesy of Westinghouse Underseas Division of Westinghouse Electric Corporation. The report was prepared by R. H. Multer, Mathematician, under the general supervision of Thorndike Saville, Jr., Chief, Research Division.

At the time of publication, Lieutenant Colonel Edward M. Willis was Director, Coastal Engineering Research Center, and Joseph M. Caldwell was Technical Director.

NOTE: Comments on this publication are invited. Discussion will be

published in the next issue of the CERC Bulletin.

This report is published under authority of Public Law 166, 79th Congress, approved July 31, 1945, as supplemented by Public Law 172, 88th Congress, approved November 7, 1963.

CONTENTS

Page

Section I. INTRODUCTION .

Section II. THE WESTINGHOUSE E. FLOWMETER.* N 2

Section III. UNIFORM VELOCITY TOWING TESTS . .i . . , , . 6

Section IV. OSCILLATING CARRIAGE TESTS - . . . - 10

Section V. WAVE TESTS . 4 '"! :6/ ,),, 6 ', 20:

Section VI. GEOMETRIC DESIGN OF PROPOSED FLOWMETER . . 24

Section VII. DETERMINATION OF VELOCITIES WITH THE FLOWMETER . . . 25

Section VIII, CONCLUSIONS 27

ILLUSTRATIONS

Figures.

1. Acoustic Water Velocity Measurement Principle . 3

-Mounting for Experimental Acoustic Flowmeter el 4

Variation of Measured Velocities with Angle between Gage. and

Flow Directions, Carriage Speed of 13.30 feet per second Comparison of Carriage-indicated Velocity with Measured

5, Normalized Directional Response of Acoustic Gage in

- Unidirectional Towing Tests . . . . , .

. ,

6. Comparison of Corrected Measured Velocities with

Carriage-indicated Velocities # , - . . , - 9

T, Setup for Oscillating Carriage, Tests, 16

8..; Comparison of First Harmonics . _A 18

Setup for Wave Tests .19

Definition Sketch . 1, 0.) 20

Comparison of Measured and Predicted Orbital Velocity

Components . W et

...

23 Definition Sketch . . K

,

A 24 L. FLOWMETER 5 . . 7 Velocity 8

ILLUSTRATIONS (Continued)

Tables Page

Harmonic Components of Velocity for a Generator Period

of 4.00 Seconds 11

Harmonic Components of Velocity for a Generator Period

of 6.00 Seconds 12

Harmonic Components of Velocity for a Generator Period

of 10.00 Seconds 13

Harmonic Components of Velocity for a Generator Period

of 12.00 Seconds 14

Harmonic Components of Velocity for a Generator Period

of 14.00 Seconds 15 iv . .

..

.

....

. . . . . V. .

...

. .

Section I. INTRODUCTION

A device which will reliably and accurately measure velocities is needed for both laboratory and field investigation of water-wave phenomena. Development of a directional gage suitable for prolonged use in the nearshore marine environment has been emphasized.

This report contains the details of an investigation which was undertaken to adapt a commercially available acoustical flowmeter for the desired directional measurements. Williams (1965)* conducted tests with a two-probe flowmeter which measured velocity along a single path. The two probes were cylindrical mounts for acoustic transducers used to measure the flow velocity between the probes. Results from Williams' study led to this investigation because the present investigator recog-nized that a zone of low velocity flow occurs behind a cylinder immersed in a flowing viscous fluid and anticipated that the measured velocity would be distorted when the angle between the measurement path was small, but would be little affected when this angle was relatively large. The

equation

Vm = Va cos 0, _{lel > a} (1)

where Vm is the measured velocity

Va is the actual water velocity

8 the angle between the velocity and measurement path

and a an angle of approximately 300

described the anticipated response. Williams' results more or less supported the hypothesis that equation (1) was correct.

The value of a in equation (1) is of considerable importance, because (as is shown in Section VI) the wake effect may be avoided by making simul-taneous measurements along multiple paths. The number of paths necessary depends on the magnitude of a. A reliable estimate of this quantity could not be obtained from Williams' data and the experimental phase of this investigation was undertaken to develop a better description of the response function.

Three types of experiments were run in the following order;

a. Unidirectional towing tests made at the David Taylor Model Basin (DTMB) (now the U. S. Naval Ship Research and Development Center).

*Williams, L. C. (1965) "An Ocean Wave Direction Gage" from Marine Sciences Instrumentation, Vol. 3 (Plenum Press, 1965).

Towing tests in which the gage was oscillated in a tank of quiescent water at CERC.

Measurements of velocities with a fixed gage and an oscillating flow at CERC.

The two tests at CERC were conducted in the 635-foot wave tank. Results of all tests are given in Sections III, IV, and V of this report. The

tests in which an oscillating carriage was used are believed to be the most reliable of the series. Initial electronic malfunction of the

equipment and ultimate failure of one of the probes which housed an acoustical transducer precluded the completion of the test program initially planned for the DTMB. The electronic difficulties had been fixed before the second and third test series at CERC. New difficulties developed in the third series of tests. These difficulties (detailed in Section V) were not with the flowmeter, but with the recording equipment and the meter's mounting apparatus.

Section II. THE WESTINGHOUSE L. E. FLOWMETER*

The L. E. Flowmeter is designed to measure the difference in travel

times of two acoustical pulses traveling simultaneously in opposite directions along a common acoustical path, and from this difference compute the average velocity along the path. Figure 1 shows the basic acoustic arrangement. Separate receiving and transmitting elements are shown only for clarity.

The time required for a sound signal to transit acoustic path A (from projector PA in probe No. 2 to hydrophone HA in probe No. 1) is

TA = c + V

The time required for a sound signal to transit acoustic path B (from projector PB in probe No. 1 to hydrophone HB in probe No. 2) is

TB =

c - V

where

c = average velocity of sound propagation in the acoustic medium in feet per second

L = acoustic path length between projector-hydrophone sets in probes 1 and 2 in feet

*Much of the information in this section is taken from material furnished by Westinghouse Electric Corporation.

L

-Receiver A

Transmitter B

Synchronizer

1

Output ( Proportional To V )

Figure 1. Acoustic Water Velocity Measurement Principle

Transmitter A Receiver B Probe Acoustic Path B 11. Probe 2 B HB

I

Acoustic Path A P A '4 A Time Comparator

or

47= 2LV

(c21 V2)

(2)

For c2 » V2, equation (2) may be reduced to the following linear equation:

V = average velocity of water along acoustic paths A and B with respect to the projector-hydrophone sets (fixed) in feet per second

If transmitters A and B, simultaneously excite projectors PA and Pg with signals of a suitable characteristic at some time To, then these

signal will appear' at receivers A and B at subsequent times

Ti F

T +T =T

+

A a -A o, t + V

T* T +, T = T + L

a c - V

The time comparator measures the difference between T TA

(i.e.,

LT)

T=

TA - TA

To +

L

o + V t-- :V t + V

with an error of only 0.047 percent When V is 1.001 feet per sec.

Equation (3) Shows that compensation must be made for the speed of sound which might vary from 4,600 to 5,200 feet per second under extreme ranges of temperature and salinity. A computing circuit within the system measures the speed of sound and corrects the output automatically for any variations in the

transmission

properties of the medium.

The necessary acoustic signals are produced by a pulsed transmitter 'which 'causes identical transducers, at opposite ends of the transmission path, to be energized simultaneously. The generated pulse shock excites

the transducers causing them to "ring". briefly at their resonant frequency.. The sound pulse is received at the opposite end of the transmission path by the opposite transducer. This is accomplished by having each trans-ducer serve first as a projector and then as a receiver. Upon receipt at

the transducer, the

resonant

frequency pulse is passed through a gated-receiver circuit which in turn drives a threshold circuit. The gated receiver eliminates any false response due to reflection or the side lobes of the acoustic signal. Since the trigger level of the threshold circuit

11

2L

v.

(3)

B o B

-3n Diameter

3/4" Diameter

21

Transducers

Figure 2. _{Mounting for Experimental Acoustic Flowmeter}

5

is so much lower than that of the signal from the receiver, the infor-mation is independent of amplitude as long as there is enough signal received to trip the threshold.

The only variable in the system is the time difference, AT, between the arrival of the two signals at their respective transducers. Since AT varies linearly with water velocity, a system that quantizes time in small increments and counts them linearly will give the best performance. This is accomplished in the system by feeding a high-frequency square wave into a digital counter. The square wave is gated into the counter only for the time beginning with the first received signal and ending with the second received signal. Therefore, a number of counts that is directly proportional to AT is read into the counter.

Section III. UNIFORM VELOCITY TOWING TESTS

Extensive towing tests were conducted at the David Taylor Model

Basin. The four-probe sensor configuration shown in Figure 2 was used

for these tests. Two of the probes were dummies mounted to obtain geo-metric similitude between the test gage and the envisioned design. In

these, and subsequent tests, the effect of changing velocity direction was necessarily obtained by rotating the gage, and the use of dummy probes was thus satisfactory.

The towing tests were conducted by mounting the test gage on the towing carriage and aligning the probes containing the transducers with the direction of carriage movement. Then, with the carriage moving at a constant speed, the test gage was turned in 5-degree increments through

an angle range of ±90 degrees. Care was taken to ensure that the gage was towed in quiet water. Velocities indicated by the gage were recorded

on a paper strip chart. Carriage speeds, furnished by DTMB equipment were 1.65, 3.34, 4.99, 6.68, 9.99, and 13.30 feet per second.

The directional response initially obtained was not as anticipated (Figure 3). The unanticipated side lobe variation was found to be independent of gage orientation or towing carriage velocity (Figure 4).

It was concluded that the observed response was the result of electronic malfunction and not a real physical characteristic of the flow. Following

completion of the tests, an electronic (static) meter calibration was made, and the results of this calibration were consistent with that obtained from the towing test data. An improperly wired circuit apparently caused the difficulties. Data from the tests were corrected in accordance with the calibration curve (Figure 4) and the results obtained were relatively satisfactory (Figure 6). Some specific results are shown in Figure 5. These results are for a probe spacing of 2 feet and a velocity range of from about 2 to 14 feet per second. The velocities for the 1.67 feet-per-second carriage velocity tests are apparently too large. Other tests at this velocity show results which are too low. Therefore, it is

be-lieved the actual difficulty is, at least partially, a limitation of the

14 13 12 10 5 4 3 2 0 90

Corrected Velocity Apparent Velocity

+ 60 30 Angle (Degrees) 0 Figure 3. Variation of Measured

Velocities with Angle between Gage and Flow Directions, Carriage Speed of 13.30 feet per second.

10 0 Tu 9

>

2 3 4 5 6 7 8 9 10 II 12 13 14

Velocity Indicated by Acoustic Gage

Figure 4, Comparison of Carriage-indicated Velocity

with Measured Velocity (Gage Angles between +15 degrees and -15 degrees were not used.)

and Carriage ft./sec. 3.34 Velocity ft./sec Symbol 1.65 _4.99 . . 6.68 9.99 ft./sec.

ft/sec.

ft./sec.

a + 13.30 4i i x

II

Mr/

IIIIIIEW

Fr

Wilniari

keiMPAbal

.

AA

Derived Calibration Curve

14 13 5 cn 4 0 3 2

0

T

-ILO 0.5 100 1.0 > 0,5 00 -90 -60 -30 0 +30 +60. 1- Left Right Angle Left Right -Cos 00 +90' -60 -30

Lift

Left. cos e Carriage 'Speed [1.65 ft./sec. 0 +30 Angle -90 -60 -30 0 +30 +60 +90 Angje

1"--.

Left t Right 1

JI,eft TRight

Carriage Speed 4.99 ft /set. Carriage Speed 9.99 ft/sec. +90 +60 +90 Right

--41

Right 1.0 0.5 00 1,01->0 0.5

< 0.5

Left , Cos -90 -60 -30 Left Cos (Left Left 00 Left 0 ,Angle Carriage Speed 3.34 ft./sec. Right Right Carriage Speed 6.68. ft/sec. 00 -90 -60 -30 0 +30 +60 +90 Angle Right Right *30 +60 Right Right

-Figure S. Normalized Directional Response of Acoustic Gage in Unidirectional Towing Tests Showing Distortion of

Velocity for Small Angles of Approach.

8 -+90 0 +60 +90 0 4-Angle Cos 9 ..Corr'440 Speed 13.30 ft/sec. 1.0 Cos 9 1 1.0 0 9 -90 -60 -30 Left

1 I 2 1 9 , i

4,5,6,7

10,11,12)3 14,15,16;17,18,19 Run

8,9

20

21,22. 1 Carriage

6.68

3.34

9.99

13.3 1.65

4.993

Speed

ft/sec.

ft/sec. ft/sec.. ft/sec..

ft/sec.

ft/sec,

Symbol' o a, .4.

.

+ ...

II

1

rig

i

OF=

id

I

,

111

x . '

3

4 5 6 7 S 9 11 112 13

Cosine of the Angle X Carriage Speed

Figure 6. Comparison of Corrected Measured Velocities with Carriage-indicated Velocities (Unidirectional Towing Tests)

12

4 3

accuracy of the DTMB meter used in measuring carriage velocity at this

low speed. No significant trends of the ratio Vm/Va were noted for small

values of 0 (defined with equation 1). In fact the data was statistically consistent with the equation

Vm

= vc

f(e) (4)

where*

f(e) cose, lel > 15°

f(0) 0.83

as is shown on Figure 6, although the correction necessitated by the electronic malfunction is not precise.

Section IV. OSCILLATING CARRIAGE TESTS

The second series of tests was conducted in CERC's 635-foot wave tank using the setup shown on Figure 7. The only motion in the fluid was that induced by the oscillating flowmeter.

The oscillatory motion of the meter is given by the equation

X(t) = 8.75 cos(wt) + 44 {1 ( 8.75 )2 2

I sin (wt)

\ 44

as may be deduced by examining Figure 7. Hence the motion consisted of a large primary sinusoidal motion and much smaller higher harmonics. The first terms of the Fourier series which represents the displacement function are:

X(t) = 43.56 + 8.75 cos(wt) + 0.44 cos(2t) - 0.001 cos(4wt) (6)

as may be determined theoretically from the displacement equation if a constant angular velocity, w, is assumed. The correctness of the equation was verified by supplemental measurements conducted as a part of the test program.

Harmonic components of velocities determined from tests with 4, 6, 10, 12, and 14-second periods are given in Tables I, II, III, IV, and V. The velocity components determined from the acoustic flowmeter data are tabulated in the columns headed COMP, while the velocity components de-termined from the displacement equation (equation 6) are tabulated in the columns headed THEOR.

(5)

* The range of 6 is being taken as -90° s 8 s 90°. In other words 8 is the minimum angle between the velocity vector and the measurement path.

10

-TABLE I

HARMONIC COMPONENTS OF VELOCITY FOR A GENERATOR PERIOD OF 4.00 SECONDS*

*The tabulated velocities are in feet per second

Angle Mean Velocity First Harmonic Second Harmonic Third Harmonic Measured Variance Residual Variance Computed Period COMP. THEOR. COMP. THEOR. COMP. THEOR. -80 -.069 2.346 2.387 .159 .235 .224 .000 9.97 .07 4.04 -70 -.106 4.355 4.701 .354 .462 .380 .000 33.70 .09 4.03 -60 -.051 6.254 6.872 .423 .675 .473 .000 71.07 .06 4.06 -50 -.057 8.174 8.835 .593 .868 .549 .000 118.36 .26 3.98 -40 -.434 9.620 10.529 .885 1.035 .595 .000 178.88 9.70 4.08 -30 -.352 11.164 11.903 .760 1.170 .634 .000 221.31 .42 3.98 -20 -.151 11.847 12.916 .867 1.269 .606 .000 249.98 .84 4.02 -10 .240 11.639 13.536 .988 1.330 .741 .000 240.73 .46 3.99 -05 .207 10.502 13.692 .774 1.346 .834 .000 194.26 .47 4.01 00 -.178 10.288 13.744 .857 1.351 .754 .000 189.58 .35 4.00 +05 -.219 10.612 13.692 .963 1.346 .828 .000 199.27 .45 3.99 +10 -.215 11.757 13.536 .922 1.330 .802 .000 248.88 .50 4.00 +20 -.174 11.681 12.916 1.012 1.269 .786 .000 244.46 .45 4.00 +30 -.162 10.911 11.903 .876 1.170 .703 .000 212.26 .34 4.00 +40 .000 9.596 10.529 .663 1.035 .670 .000 165.31 1.33 4.06 +50 -.029 8.039 8.835 .698 .868 .689 .000 115.01 .17 4.00 +60 -.001 6.297 6.872 .554 .675 .478 .000 70.53 .16 4.01 +70 -.062 4.481 4.701 .361 .462 .346 .000 35.44 .11 4.00 +80 -.073 2.439 2.387 .210 .235 .269 .000 10.67 .04 4.01 +90 -.009 .215 .000 .078 .000 .061 .000 .16 .06 3.98

TABLE

II

HARMONIC COMPONENTS OF VELOCITY FOR A GENERATOR PERIOD OF

6.00 SECONDS*

*The tabulated velocities are In feet per second.

Mean Angle Velocity First Harmonic Second Harmonic Third 'Harmonic Measured Variance Residual Variance Computed Period COMP. THEOR, COMP. THEOR COMP. THEOR: 0.80 -1.865 1.943 1.591 ,171 .156 .055 28,22 c13 6,03 -70 .3.079 3.361 3.134 *295 .308

all

.000 79.76 .11 6,00' -60 -4.346 4.652 4,581 ,414 .450 .069 .D00 155.88 .16 6.01 .50 .5.315 5.807 5.890 .559 .579 .154 .1000 236,86 .14 5.99 -40 .6.452 6.920, 7,015 .688, .690 .189 :000 344.18 .22 6.01 -30 -7.189 7.714-7.935 .712 .7801 *110 .000 425.50 .29 6.00 -20 -7,770 8.120 8.610 ,754 .846 .1S4 .000 488.03 ,35 5.99 .10 .,7.361 7.970 9.024 .781 ,887 .174 ,.000 447.69 .38 5.99 -05 -6,597 7,222 9.128 ,699 .897 ,136

!OH

365.46 .73 6.03 00 .6.838 7.113 9.163 .672 .901 .261 ,000 385.67 2.59 6,09 405

-6372

74304 '9,128 .767 .897 .303 .000 355.24 :41 6.02 +10 u7.255 7.722 9.024

884

.887 .249 .000 440.21

A8

,6.06 +20 -7.480. 7.974 8,610

826

,846 ,211 .000 466.24 :89 6.06 430 -.6.825 7,523 7.935 .712 .780 .115 :000 394:59 .29 6.02 +40 -6.075 6.711 7.019 .609 .690 .135 ,000 310.78 ,.26 5,99 450 .25.231 5.706 5.890 .524 .579 ,155 .000 229.03 .16 6,01 +60 .254 4,420 4.581 .4191 :450 .081 ,000 52.38 .15 6,01 +70 .120 3.134' 3:134 .281 .308 .101 .000 26.20 :12 6.02 +80 .048 1,673 1.591 .151 .156 ,069 .000 7.81 .17 6.02 +90 .026 .169, .000 .065 .000 .095 :.000J ,26 .15 6.08

HARMONIC COMPONENTS OF VELOCITY FOR A GENERATOR PERIOD OF 10.00 SECONDS*

-_

-The tabulated velocities are

in

feet per second,

TABLE III Angle Mean

yelocitz

First Harmonic Second Harmonic Third Harmonic Measured Variance Residual Variance Computed Period COMP.. THEOR. comp. THEOR. COMP,

THEM,

,-. ... -80 -.126 1.169 ..955 .118 .094 ,072 .000 12.69 .16 10.02

.7o,

.._ .60 _ -,.193 _-,187 2.054 2,785 1,880 2.749 ,175 .199 .185 :270 .077 ,151 ,000 .000 39.65 69.18 .25 ;27 10.32 10.00 :.-51? *.136 3.628 3.534 ,304 4347 T152 4000 115.43 ,35 9,98 illi, *30 *.194 -.215 4,269 4.756 4.212 4.761 .358 .366 ,4114 ,468 .177 _%142 .0001 .000 163,85 199.28 .4a .38

96

9.98. *20 ,.,152 4.948 5,166 .401 4508 ,120 .000 216.66 .45 10.08 "."1.

w

-10. -.121 4.828 5.414 ,381 .532 .155 .000 210.57 .67 10.04 -05 -.174 4.494 5.477 .372 4538 ,205 ,000 178.85 .59 10.04 pp ..227 4.503 5.498 .383 .540 .163 ,000 180.24 .58 10.00 +05 ...246 4.565 5.477 ,418 .538 .197 .000 184.56 .60 10.00 +10 --.231 4.851 5.414 ,461 .532 .132 .000 212,50 .62 10.00 +20 -.151 4,913 5.166 .469 .508 .134 ,000 215,63 ;56 9.98 +30 0117S 4.614 4.761 .371 .468 ,176 ,000 188.98 .55 .9,98 +40. -.153 4.118 4.212 .336 ,414 ,181 .000 148.92 ,36 10,02 +50 ,247 3.524 3.534' ,327 .347 .110 ,000I 114,92 ,56 10.00 +60 -.193 2.736 2.749 .224 .270' ,185 ,000 67.73 .29 9,98 +70 .174 2.084 1.8B0 .157 .185 ,077 ,000 38.96 .26 10.02 +80 ...191 1.140 .955 ,,136 ,094 ..074 ,000

12.34.21

9.98 +90 .--.155 .212 ,000 .063 .000, ,027 ,000 1,17 :53 9.90

* The tabulated velocities are in feet per second.

TABLE IV

HARMONIC COMPONENTS OF VELOCITY FOR A GENERATOR PERIOD OF 12.00

SECONDS* Angle Mean Velocity First Harmonic Second Harmonic Third Harmonic Measured Variance Residual Variance Computed Period COMP.

TOR.

COMP. THEOR. COMP. THEOR. -90 .033 .020 .000 .021 .000 .009 .000 .06 .03 11.83 -80 -.055 1.078 .796 .106 .078 .057 .000 12.58 .18 11.91 -70 -.060 1.852 1.567 .126 .154 .073 .000 35.75 .28 11.95 -60 -.058 2.518 2.291 .193 .225 .172 .000 66.81 .25 11.96 -50 -.043 3.053 2.945 .228 .289 .162 .000 98.17 .40 11.94 -40 -.112 3.652 3.510 .311 .345 .139 .000 140.81 .45 11.92 -30 -.110 4.034 3.968 .322 .390 .189 .000 172.58 .66 12.02 -20 -.105 4.292 4.305 .344 .423 .174 .000 195.57 1.00 11.88 -10 -.120 4.029 4.512 .317 .443 .242 .000 177.65 1.60 12.37 -05 00 -.038 -.141 3.842 3.886 4.564 4.581 .297 .333 .449 .450 .174 .230 .000 .000 156.86

16134

. .58 .48 11.97 11.95 +05 -.147 3.937 4.564 .310 .449 .156 .000 163.73 .48 11.89 +10 -.143 4.059 4.512 .352 .443 .229 .000 180.03 1.05 12.19

BAD DATA FOR +20 degrees ₊₃₀

-.058 3.911 3.968 .317 .390 .191 .000 165.42 1.66 12.18 +40 -.153 3.586 3.510 .280 .345 .148 .000 136.46 .54 11.96 +50 -.115 2.999 2.945 .254 .289 .178 .000 96.42 .40 12.01 +60 -.158 2.408 2.291 .176 .225 .128 .000 63.82 .25 12.36 +70 -.114 1.819 1.567 .126 .154 .093 .000

35.15.34

11.87 -. '

TABLE V

HARMONIC COMPONENTS OF VELOCITY FOR A GENERATOR PERIOD OF

14.00 SECONDS*

* The tabulated velocities

are in feet per second.

Angle Mean Velocity First Harmonic Second Harmonic Third Harmonic Measured Variance Residual Variance Computed Period COMP THEOR. COMP. THEOR. COMP. THEOR. -90 .011 .042 .000 .035 .000 .033 .000 .13 .08 14.15 -80 .003 .907 .682 .095 .067 .064 .000 10.49 .26 13.91 -70 -.062 1.592 1.343 .120 .132 .070 .000 31.99 .33 14.06 -60 -.047 2.227 1.963 .168 .193 .111 .000 62.09 .37 14.06 -50 -.091 2.656 2.524 .172 .248 .167 .000 88.72 .46 14.17 -40 -.116 3.139 3.008 .261 .296 .171 .000 123.20 .48 14.05 -30 -.116 3.531 3.401 .292 .334 .118 .000 154.57 .68 13.97 -20 -.060 3.683 3.690 .303 .363 .121 .000 168.60 .52 14.05 -10 -.080 3.610 3.867 .306 .380 .154 .000 162.29 .73 13.98 -05 -.088 3.396 3.912 .260 .384 .148 .000 143.25 .63 14.01 00 -.117 3.404 3.927 .290 .386 .142 .000 144.63 .68 13.98 +05 -.057 3.409 3.912 .265 .384 .148 .000 149.13 .94 14.03 +10 -.097 3.559 3.867 .277 .380 .164 .000 158.65 1.01 14.09 +20 -.120 3.663 3.690 .308 .363 .142 .000 165.59 .49 14.02 +30 -.096 3.427 3.401 .318 .334 .137 .000 147.87 .88 14.03 +40 -.183 3.097 3.008 .264 .296 .148 .000 119.69 .49 13.97 +50 -.170 2.610 2.524 .196 .248 .188 .000 85.57 .34 14.15 +60 -.124 2.188 1.963 .190 .193 .083 .000 59.95 .35 14.08 +70 -.183 1.570 1.343 .134 .132 .092 .000 31.69 .34 14.01 +80 -.116 .908 .682 .115 .067 .054 .000 10.90 .25 14.02 +90 -.095 .136 .000 .054 .000 .042 .000 1.03 .51 14.02

Acoustical Meter.

I 2' Eit Towing Carriage Dry Bulkhead kJ. Stroke =17.5'-.44 Vertical Bulkhead

/Wave Generator

Figure 7.

Setup for Oscillating Carriage Tests

Generator Fly Wheel

'

fl

K=1

-A plot of the first harmonic of the velocity determined from equa-tion (6) versus the first harmonic of the velocity determined from the acoustic-flowmeter data is given in Figure 8. Only the data for angles of not less than 200 to the oscillatory direction are plotted. A feature

of this graph is that the zero amplitude of the first harmonic of acoustic-flowmeter velocity does not coincide with the zero of the theoretical first harmonic. This difference is explained as a property of the digital-to-analog conversion which was made.

The acoustic-flowmeter velocity harmonics given in Tables I through V are those for which E(w), defined by

2

((ti)1 = Vk A0 4 (W)le Wtk (55 )

k=1 j=1

was minimized, wherein the equation w

is

an undetermined angular velocity, the Vk are measured velocities, and tk are the corresponding times at which measurements were made. Minimizing E(w) is a nonlinear problem. It was found that a quite reasonable approximation could be obtained by minimizing E(w) for a short sequence of equally spaced wn's centered around the nominal angular velocity of the wave generator. Trial com-putations indicated that the digital computer program used for this purpose produced results which were accurate to more than 4 decimal places.

Data reduction (digitizing) was accomplished on the CERC Auto-Trol Model 3400 digitizer. Data points were taken at equal increments of (St. The sampling increment was chosen so that more than 25 data points would be sampled from each period and a record about 5 periods long was sampled in all cases. _{Hence, reasonably accurate estimates of the velocity}

components were obtained.

The entries in the columns titled variance extept in Table II are reasonably accurate estimates of the sum*

2

.= ( _{VK -} )

17 and the average velocity _V ~ CY.

The column titled residual variance contains, corresponding valueS of the computed minimum of E(w). _{It is significant that E(w) is almost} always. less than 1/2 percent of S. - i.e. almost all of the measured velocity variance is explained by the Fourier sum indicated in equation

(6). _{Additional computations have shown that the residual fluctuation}

M

1

-actually' S = (1/K )

2

and the average velocity V O.

X=1 + ( - -S -*

11

8

6 :5 3 2 0

Vellocilv Computed from Displacement Functio

Figure 8.

Comparison of First fth.trilOtlics

18 , .. .

1

1 II

-1 I I 1 - , , _ I II -.-_-. , . x x i 1 1 , 1 ,PIERIOD,

4.0

6.0

110.0 112.0 14.0 , LEGEND SYMBOL o a, a + I - , 1 .ix 1. 4...,+ + I I -1 7----II '

r

,S

I ,

i

__---.,___ r -. I

/

---

+I-t

2

3

4

5

16 7 'R

in

it

1; 12 9 4

/

//0

9

Figure 9. _{Setup for Wave Tests (resembles envisioned field des'gn)}

is relatively uniformly distributed in the frequency domain. Hence, the distribution of kinetic energy is entirely consistent with the physical

situation.

Section V. WAVE TESTS

In the third sequence of tests, the acoustic flowmeter was mounted on the bottom of the 635-foot wave tank as shown in Figure 9. Directional response of the meter in a wave motion differs from its response in the previous cases in that attenuation and phase shifting of the fluctuating velocity must now be considered. The attenuation and phase shift of a sinusoidal velocity is shown by the following calculation

Figure 10. Definition Sketch

We assume a velocity V given by

V = A cos (kx - wt + TO From Figure 10 VI = V cos(a-0 and V2 = V cos(A) 20 X

In the case of interest the dimensionless parameter

- where St is the travel time of an acoustical impulse along the measure-ment path and C the acoustical impulse's celerity - is extremely small. Hence the acoustical meter essentially averages the velocity over the distance 6s at time t. The average velocity, V1, is

6s äx

V

= 1

S

V dx A cos(a-e) cos(kx - wt

+ T)

dx 1 T.s. 1 1 (5x1 where ax1 = ós cos(a-e) or on integrating

Vi = A cos(a-e) [al cos(wt-T) + bl sin(wt-T)]

where al E sin(k6x2) / k6x1 and and similarly 21 bl E [1 - COS(k6X1)] The attenuation 2 . 2 ) p1 = (al + _b1 and -1 n1 = - sin / k6X1

and phase shift

2 Sx, n1 are then dx1 (7a) (7b) = sin k kox1 (131411) = 2 That is v1(t) = p A cos(a-e) cos [wt - (T

+1)]

(8a) A St) k

22

v2(t) = _p2A cOS(&) cos [wt -

(T + n2)]

(8b)

where the equations for the parameters p2 and

n2 are obtained from the

equations for

pl

and n1 by replacing the subscript 1 with the subscript

2.

Equations 8a and 8b could be considered simultaneous nonlinear equations in the two unknowns 6 and A if k is known. Experimental determinations of k in laboratory investigations of unidirectional oscillating flow can be made with the meter, but this is beyond the scope of our present inter-est in field instrumentation.

In field investigations, the situation is more complex. We note, however, if k6x is relatively small, then sin (k5x/2)/(k(5x/2) is very nearly 1 and the attenuation is unimportant. Equations 8a and 8b and

linear wave theory can be used to estimate the magnitude of the attenua-tion error which would be expected in a specific case.

The response of the meter to wave flows was investigated for the condition indicated on Figure 11. These cases involved relatively small amplitude waves, and the flowmeter data indicated that the energy of the orbital velocity (as determined by the analysis indicated in Section IV) was almost entirely at the frequency of the generator motion. A comparison of the measured amplitude of the orbital velocity with that predicted by linear wave theory is made on Figure 11.

The significant feature of this graph is that the curve again does not pass through the origin. The most plausible explanation of this

phenomena is that the digital-to-analog converter had a nonlinear response at relatively low voltage outputs.

1.2 1 .0

0.8

0.6

x

0.4

0.2

0.0

0 0.2 0.4 0.6

0.8

1 0

SIN(a) x Umax (Linear Wave Theory)

Figure ll. Comparison of Measured and Predicted Orbital Velocity Components

23 12

i

.=

t

a_ L.7_

a Values are 10°,20°,30';

A

,

a 1

4050°,60°7

V

y"

Z"

X

7

X

V

V

T(Sec) H(Ft) D(Ft.) Symbol

2.6

1.78

6.2

0

4.0

1.37

6.1

8.0

0.71

6.1

I I,

Section VI. GEOMETRIC DESIGN OF PROPOSED FLOWMETER

A flowmeter with three probes located at the nodes of an equilateral

triangle is recommended as being appropriate. This geometric design is suggested because the experimental evidence indicates that as long as the

angle between the velocity vector and the measurement path exceeded 25°, the function which related the measured velocity to the actual velocity, is strictly monotonic - as a function of angle; and because of the

following:

Figure 12. Definition Sketch

Proposition:

If

PiP2P3

is an equilateral triangle and 0 the center of the

in-scribing circle, then any line through 0 intersects the triangle at two

points A and B and any resulting triangle _ABPk has a minimum angle of 716.

Proof, without loss of generality, assumes A is on the line

P3P1 and B is on the line P1P2. Evidently the hypothesized

triangle is ABP1 only as long as 716

< e <

712; but in this

case the range of the angle BAP1 is clearly 716 to 7/2 the angle

APO =

7/3 and

/ABP1 =

7 -

(/AP113 + /BAP].)

hence the range of the angle /ABP1 is 7/6 to 712 also.

In other words, if we measure the velocity components along each of the sides of the equilateral triangle there are (generally unique) two sides along which the velocity components are at angles of at least 300 to the velocity vector.

Our experimental evidence indicates that equation (1) is essentially correct for a=25°. Hence, if velocity measurements are made along the sides of an equilateral triangle there are always at least two measured velocity components which are at an angle of 30° to the velocity vector. These components are, therefore (essentially) undistorted, averages - in time - of the velocities along the paths, which is the reason for assert-ing that this geometrical probe configuration is appropriate.

Section VII. DETERMINATION OF VELOCITIES WITH THE FLOWMETER

If we let v be the magnitude of the velocity, then, the measured velocity components will be

v. = v cos (13-a.)

j

= 1,2,3

(9)

where

a is the orientation angle of the velocity vector

and

a is the orientation angle of the jth path

Designating any two paths by the indices 1 and

2,

equation (9) can be used to derive the following two equations:

cos (3 cos

a1 + v sin IS sin

a1 = v1

cos

a cos a2

+ v sin 13 sin a2 = v2

On solving these two equations we find

cos 13 =

sin

(a2

-

al)

v2 cos

a1

-

v1 cos a2 sin 13 =

sin

(a2 - al)

From which it follows that

2 2 2 2 v sin

(a1-a2)

=

vi

v2 2v1v2 cos

(a1-a2)

(12)

vI

sin

a2 - v2

sin a1 25 (10) v

-Next, let us suppose that

v2 is distorted, that is, that the actual measured component is

and therefore

2 2Av1v2 cos (al-a2) s Xv2

in that cos (a1-a2) ± 1/2

Hence equation (13) implies

2 .

2. 2

V sin 2 2

m (al-a2) v2 sin (al-a2) + (X - X) v2

Now the quantity X2-X is negative for 0 < X < 1 so that we may infer

2 2

vm < v

from equation (14).

Hence we conclude that the pair of indices which indicate the greatest velocity are undistorted.

I

v1

26

(13)

u = v2 (1 - A) 0 < X < 1

as would be consistent with experimental data. Replacing

v2 with u in equation (13) leads to

2

.2

2 2

l-a2)

2 2 2

vm sin (a1-a2) =

Iv

sin - 2Xv2 +

X v2 (a

+ 2Xv1v2cos (a1-a2)1

where

vm is the measured speed.

Since

v2 is distorted, the angle between the measurement path along

which

v2 is measured and the velocity vector must be less than

7/6

and the angle between the measurement path along which vl is measured and

the velocity vector must be at least 7/3. Hence from equation (9)

Section VIII. CONCLUSIONS

As tested, the Westinghouse Acoustical Flowmeter produced data on orbital velocity, and current magnitude and direction, which _{are judged} to be adequate for many oceanographic investigations. _{The primary} limita-tion of the system is the effect of probe spacing _{on the accuracy of the}

system. _{This factor precludes miniaturization of the system to the extent}

necessary for use in many small-scale laboratory studies. Advances in the development of electronic and acoustic components would be required _to overcome these limitations.

Satisfactory results may be obtained with the meter only if _velocity components are measured along multiple paths. _{Because of the wake effect,} measurements along more than two paths must be made to obtain the two

horizontal components of velocity. _{Measurements along the three sides} of an equilateral triangle would be adequate as is shown in Section VI. The correct magnitude and direction of the velocity _{may be found by using} the method developed in Section VII.

UNCLASSIFIED

Dr),""...1473

AAAAA CE Do 01,1m1471. I JAN 04. VIMICH II

01110LKTII VOIR ARMYult. _UNCLASSIFIED

Security Classification

--1DOCUMENT

CONTROL DATA . R & D

(Security classification of title, body of abstract and indexing annotation must be, entered when the overall re.port I. classified) I. ORIGINATING ACTIVITY (Corporate author)

Coastal Engineering Research Center (CERC) Corps of Engineers, Department of the Army Washington, D. C. 20016

24. REPORT SECURITY CLASSIFICATION UNCLASSIFIED

2b. GROUP

3. REPORT TITLE

MEASURING DIRECTIONAL VELOCITY IN WATER WAVES WITH AN ACOUSTIC FLOWMETER 4 DEsCRwPT VE NOTES (Type of report and inclusive date.)

5. Au THORIS1 (First neon, middle initial, last name) Multer, IL H.

6. REPORT DATE April 1970

7.. TOTAL NO. OF PA,ES 27

7b. NO OF OFFS 0 .a. CONTRACT OR GRANT NO.

b. PROJEC T NO.

,

d.

64. ORIGINATOR, REPORT NOM.BERIS,

Technical Memorandum No. 31

Ob. OTHER REPORT NOIS1 (A, other numbeire that may be auslgned this ',ore)

10. DISTRIBUTION STATEMENT

This document has been approved for public release and sale; its distribution is unlimited.

11 SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

13 ABSTRACT

This report contains the technical details of an investigation which was undertaken to adapt an acoustical flowmeter to a device for measuring velocities in water-wave phenomena. _{The flowmeter studied was designed to measure the} difference in travel times of two acoustical pulses traveling simultaneously in opposite directions along a common path. _{Because of viscous effects, a zone of}

low velocity flow occurs behind each probe and the measured velocity is somewhat less than the actual velocity when the angle between the acous..ical path and velocity vector, 0, is small. _{When this angle is relatively large, the wake has}

little effect on the velocity and

Vmeasured = Vactual x cos°

It is shown in this report that the wake effect may be eliminated by making simultaneous measurements along multiple paths.

UNCLASSIFIED Security Classification UNCLASSIFIED SecurityClassification ¶4 , KEYWORDS LINK A LINK E3 LINK C

ROLE WT ROLE WT ROLE WT

, Dynamic Oceanography Water waves Wave-direction sensor Acoustic Flowmeter

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