HYDRO- 0G
AEIODYNAMISK
QRAT...RIUM
HYDRO- AND ERODYNAMICS LABORATORY
A
Lyngby - Denmarklab.
y.
Scheepsbouwkunde
Technkche Hogeschool
Deift
COMMISSION:STER FARIMAGSGADE 3 COPENHAGEN
DENMARK
Hydrodynamics
Section
Report No. Hy-3
May 1964
A Five Hole Spherical
Pitot Tube for Three Dimensional
Wake Measurement
BY
HYDRO-. 0G AERODYNAMISK LABORATORIUM
is a self-supporting institution, established to carry out experiments for industry and to conduct research in the fields of
Hydro- and Aerodynamics. According to its by-laws, confirmed by His Majesty the King of Denmark, it is governed by a council of eleven members, six of which are elected by the Danish Government and by research organizations, ard five by the shipbuilding industry.
Research reports are published in English in two series: Series Hy (blue) from the Hydrodynamics Section and Series A (green) from the Aerodynamics Section.
The reports are on sale through the Danish Technical Press at the prices stated below. Research institutions within the fields of Hydro- and Aerodynamics and public technical libraries may, however, as a rule obtain the reports free of :harge on application to the Laboratory.
The views expressed in the reports are those of the individual authors
Series Hy:
No.: Author: Title: Price: D. Kr.
Hy-1 PROHASKA. C. W. Analysis of Ship Model Experiments
and Prediction of Ship Performance 5,00
Hy-2 PROHASKA, C. W. Trial Trip Analysis for Six Sister Ships 6,00
Hy-3 lLOVl, V. A Five Hole Spherical Pitot Tube for 6,00
Three Dimensional Wake Measurements
Hy-4 STR0M-TEjSEN, j. The HyA ALGOL-Programme for Analysis
of Open Water Propeller Test
6,00
Hy-5 ABKOWITZ, M. A. Lectures on Ship Hydrodynamics - 20,00
Steering and Marioeuvrability
Series A:
No.: Author: Title: Prtce: D. Kr.
A-1 TEJLGARD JENSEN, A. An Experimental Analysis of a Pebble Bed Heat
HYDRO- O
AERODYNAMISK
LABORATORIUM
Lyngby - Denmark
A Five Hole Spherical Pitot Tube for
Three Dimensional Wake Measurement
by
V
V. Silovic
Hydrodynamics Department
ii Table of Contents Page Abstract i Introduction - 1 Theoretical Background 1
Description of the Instrument 2
Calibration 6
Measurement of the Wake Pattern 8
Accuracy
n
Concluding Remarks 13
Acknowledgment 13
References 13
Appendix A: Derivation for the Ideal Case 14
Appendix B: Block Diagram and ALGOL Source Program of the Five Hole Spherical
iii
Notation x)
a Angle between the vertical and radius r.
b Pressure at bottom hole.
beta Angle between Vh and the direction of motion.
c Pressure at centre hole.
CG Cos of the angle between Vz and T.
N Component of the velocity of flow in the direction of motion.
p Pressure at port hole.
pa Pressure at point a.
pb Pressure at point b.
pc Pressure at point c.
Po Pressure in the undisturbed flow.
pi Pressure at any given point.
R Distance from the axis of symmetry of the probe to the point of rotation of the instrument.
r Distance of the spherical head from the propeller axis.
Ra Radial component of the velocity of flow.
s Pressure at starboard hole.
SG Sine of the angle between Vz and T
V Velocity of flow.
Vh Component of V in the plane, defined by the three measuring points.
Vm Speed of the model.
Vx Identical to N.
Vy Component of V perpendicular to Vx and Vz.
Vz Component of the velocity o± flow in the z direction, i.e. in the plane of symmetry of the instrument, perpendicular to the axis of the probe and through the point c.
T Tangential component of V perpendicular to Ra and N.
t Pressure at top hole.
Angular distance between the centre hole c and each of the side holes.
(3 Angular distance between the direction of V and the point c. Angular distance between the direction of Vh and the point c.
q Water density.
x) The list does not include all the identifiers used in the
Abstract
Theoretical and practical considerations for the measurement
of three dimensional wake using a five hole spherical pitot tube
are given. The construction and testing of the instrument is
dis-cussed. Results and diagrams of the calibration and wake
measure-ment on a model are included.
Introduction
The calculation of thrust and torque fluctuation gains
im-portance as ships get bigger and faster and propeller induced
vibrations become more frequent and severe. The non-uniform flow
pattern through the propeller disc behind the ship generates different
additional forces at different angular positions of the propeller
and therefore different thrust and torque.
Many devices have been developed at different laboratories
to measure flow patterns (see reference i). The five hole spherical
pitot tube, as described by Pien (Ref. 2), makes it possible to
determine the flow in two perpendicular planes in a simple and
con-venient way. Por this reason a five hole spherical pitot tube was
constructed and tested at the Hydra- og Aerodynainisk Laboratorium, (HYA)
and this report gives a description of what has been done so far.
Theoretical Background
The pressure distribution over the surface of a sphere in a
uniform flow has been defined by classical hydrodynamics. The
possibility of using the spherical pitot tube to determine the flow,
is based on this theory.
The distribution of pressure over the surface of a sphere
immersed in a stream is given by the following expression:
PPo
2 - sin2 (i)
Prom equation (i) Pien has derived expressions by means of
which the component vector velocity of the stream can be computed
if the pressures at three points on a certain sphere and in the
-2-PC - pb
-
(cos2/h
-cos2((h -co)
Vh2-pa-pb
sin2oO
2 pc-pa-pb -
1-cos 2(
tan 2h
h can be calculated from equation (3), ii the pressures at
tl-ie three points a, b and measured. Vh can be calculated by substituting
i3h in equation (2), and thus the vector velocity in that plane is defined.
With five points instead of three, distributed in a cross shape as shown
in fig. 1, the velocity vector of flow can be completely determined.
9,8 9
Fig. 1. Manufacturing drawing of the spherical head
The equations (i), (2) and () are in their strict sense valid
only for the ideal case, dealing with a sphere in an ideal fluid.
In reality inaccuracies may be introduced during manuiacture of the
head, caused by small departures from an exact spherical shape.
Further departures from the ideal case are due to the finite area
of holes and to viscous effects.
Equations (2) and (3) still apply in principle, but departures
from this ideal case must be found by calibration.
Description of the Instrument
The construction of a five hole spherical pitot tube gives
rise to certain problems. The flow varies over the propeller disc
so the spherical head should be small enough that the flow in the
small area defined by the holes can be treated as uniform. The
the same time it should not influence the flow around the model.
Even less may the supporting part influence the flow around the
spherical head.
Fig. 2. Spherical Pitot Tube mounted on the model.
-3-To determine the non-uniform wake behind the model one has
to be able to position the spherical head anywhere in the propeller
disc.
For the calibration it is necessary to be able to turn the instrument in the two perpendicular planes defined by the measuring
holes, i.e. vertical and horizontal. In this way it is possible to
obtain flow to the head of the instrument from different angles.
Figures 1, 2, 3 and 4 illustrate the five hole spherical
pitot tube constructed at HyA. The spherical head is made as
small as 10 mm in diameter, with five holes, .6 mm diameter each,
connected by brass and plastic tubing to the manometer board.
The side holes are spaced 200 angular distance from the central
hole, since Pien finds this to be the optimum distance.
Much care has been taken in the manufacture of the spherical
head in order to avoid unnecessary inaccuracies. This is very
4
Fig. 3. The position in the propeller disc is set on the ring.
The construction chosen for the supporting part of the
instrument is a simple one, but has been found to be satisfactory
and convenient to handle. This is true for the calibration and for
actual measurements taken on the model.
The ring and radial arm on the top of the instrument (Pig. 3)
enables the movement of the spherical head to be followed, and
makes it possible to fix it anywhere in the propeller disc.
As has been mentioned previously the five holes for measuring the pressure on the spherical head are connected by tubes to the
manometer board. The latter consists of five glass tubes where the height of the water column, indicating the pressure, can be read. Furthermore, all the tops of the tubes are connected to a vacuum
chamber. This enables the datum water level in the glass tubes to
c-c
-5
o
-6--Owing to the small diameters of the holes in the spherical
head, it takes several runs of' the carriage to build up the water
levels in the manometers to their equilibrium position (see Fig. 5).
Three to four runs have been found adequate.
Fig. 5. Equilibrium position ¿o versus number of runs
20 o (rrnW) 60 o 2 4 6 NUMBER 0F RUNS
A special clamp enables the rubber tubes joining the
pilot head to the manometer board to be opened and closed
simul-taneously at the beginning and end of each run. Final readings
can thus be taken at leisure when equilibrium has been attained.
Calibration
As mentioned before, the real case of a spherical pitot
tube differs from the theoretical sphere in an ideal fluid. Pirst
of' all there are small deviations from a pure spherical shape. Also
the holes have finite size. Therefore the calibration is necessary
to define the relation between the flow and manometer readings.
Calibration is performed by rimning the instrument under the
carriage in open water. The first step is to find the zero position,
i.e0 the position in which the stagnation point coincides with the
centre hole. The instrument is run at a certain speed and its
position is adjusted until the pressure level at all four side holes
are the same. The pressure at the central hole will be somewhat
higher, depending on the speed.
To ensure that the top and bottom holes are in the vertical
plane, a special check has to be made. The head is turned to one
side in the horizontal plane, so that the axis of symmetry of the
head and the d.irecton of motion make a certain angle. The pressure
at the two holes under consideration should be the same. If it is
not, the head has to be turned about its axis until the pressure is
40 30 20 10 o _300 _200 W_100 o (3 loo 200 4.0 3.0 2 1.0 o 300
\
CALIBRATION DIAGRAM.
/
' /
II
/
/
c3.
...
c 4...
c= 5...
c=6....
c:7...
(tb)/(2xctb)
(bt )/(2xct
(cp)/V2
cs Vt2
-\'
\
"N.,. O'"N
b)
//
/
\
\
',7
'S c=5 '\..c=7
ç = 6/'
/'c:8
\\\
N
\
/
//
=3\\///
c = 4-8
From the zero position, the instrument was turned to port,
starboard, top and bottom in steps of 30 up to
3Q0
The speed of the carriage was about 2 rn/sec. The pressures at all five holes and.the speed were recorded.
The values (s-p)/(2c_s_p) and. (t-b)/(2c-t-b) which correspond
to expression (3) on page 2 were calculated and plotted against
angle (r as shown in the calibration diagram given in Fig. 6. The
2 2 2
2.
values (c-t)/V , (c-b)/V , (c-s)/V and (c-p)/V in the same diagram
correspond to expression (2).
An additional check of the curves was made at a speed of 3 a/sec, but no ifference was perceptible.
The fact that the calibration curves differ from one another
is due to various very small manufacturing inaccuracies.
Measurement of the Wake Pattern
The instrument is mounted on the model so that the axis of
symmetry of the head is in the direction of motion of the model.
Measurements are made in the propeller disc for a model
speed corresponding to the design ship speed.
Usually a wake survey will be made for a constant radius
and changing angular position round the disc. From consideration
of the calculation of thrust and torque fluctuations this process
should be repeated for at least three radii. The number of measured
points depends on the accuracy with which the flow pattern has to be
defined, and also on the pattern itself. The top and bottom regions
are usually most difficult to define, and more points should be
in-cluded in these areas to give a good picture of the velocity gradient.
The speed of the model and the pressures are recorded as
before. But now the calculation process is reversed. The expressions
(s-p)/(2c-s-p) and (t-b)/(2c-±-b) are calculated from the measured
water heights, and the calibration diagram then entered to find the
angles . Values of (c-p)/V2 or (c-s)/V2 or (c-t)/V2 or (c-b)/V2
are then read for the ( angles thus obtained, and the speed
com-ponents V in the two perpendicular planes calculated..
The components of speed in the perpendicular planes are now again each resolved into two further components, one in the direction
of undisturbed flow, defined as the normal component, and a second
c.Q HYA_Gier iyngby-Denmark
Data Sheet for the Calculation of Veloerty from Measurements made with a 5801e Pitot Tube
page 1 2 4 5 6 7 8 9 Point Speed V Poe.in the Pr.dts Measured presures alpha r radius ç centre p port staro.d b bottom t top sec/rn degrees mm rr.' i im W an 1. rn1 R 39 0.5622
/5
//0
/46.0 /640
/20.0 /390 /2 S
4'o 0.5623 30 /10 799.0 /670//70 /48.5
/34a5 4/ 05622 45 1/0/50.0 /670
//70
/63.0 /22.0 42 0562/ 60//0
/5'. 0 /65.0
/20.0 /790
//0 0 +3 05623 75//0
/575 /593
/26.0 /870
/02.3 + 4' 0.5622 90//0
/62.0 /530 /3+0 /92.0
995 45 0562+ /05//0
/6.8 /476'
/470 /92.5
9.0
46 05627 /20
//0
/690 /442
7+6.0 /9/0
/00.5 47 05421k/35
I/O /70.0/40.0 /602 /990
/02.0+8 0562# /50
/10/690 /370 /5/0 /99.5
/0/2
49 0.5627 /65 I/O/40.0 /40.0 /35.0 /+7..6
/30.0,50 05628 /90
I/O/39.0 /370
/370 /370 7387
5/ 0.5632 /70
/10739.2 /379
/36.3/379 /380
52 05627 225
//0
/690 /45.5
7+3 0 /85.0 703.0 0 HyA-GIER Lyngby-DenmarkData Sheet for the Calculation of Velocity
from Measurements made with a 5 Hole Pitot rube
Page
Ordered for the Ship type Model No,
Position
Rake snit
"7
rn/sec knots mm mm
¿/cENc/47f 571JY
C0.4S7CiÇ'6/0/
Model speed Corresponding ehip Distance d Shaft angle Radius R The aboTe to be offioe.
Bp.
/800
1f
Measurement ndicu1ar
IRakeangle
thsart angle//g
55
0.To be Date of test Water Comments:
\ I
\,j+a,
filled -. .500 in-by the drawing Signature:
.
the test
filled in during
Day Month Year
temperature
Symmetry plane of the instruis.
22 R z ngie alp55 -/ /963 /2./ 1 2 3 4 5 6 7 8 9 Point no. Speed V
Pos.th the Pr. disc
Measured preesu.res alpha r radius o centre p port 8 etarbd. b bottom t top sec/m degrees mm mm W mm W mm W min W mm W / 0563/ 0 60 I/O 4
/099 /099 /025 //ó. S
2 05624 90 60 1/2.0/140 /03.0 /190
9.0
305622 /80
60/060 /06.0 /03.0
/02.0. /70.0 P 0.56/7 / 65 60/05,0 /060 /0/5
/00.0 //0. O5
0.56/4'/50
60/06/ /090
998/06/ /03.2
6056/8 /25
60/090 ///.0
96.5/750
9t0
70.56/2 /20
60/095 1/20
95.6 /2/0
7OComputer center commente:
Model no. 6101
Model speed
1.800 rn/sec
Corresponding ship speed 11.5
PITI0N 0f MEASUREMEWr Distance fr the aft.pp. 55.00 me Shaft angle 0. OiAlegrees knots Rake angle 0. OQlegrees Radius R 500 sin
In the table bellow the actual measured and the cceiputed results are giwen:
1m is the model speed
Vr Is the radial component of velocity and is positive if outwards from the centre
of propeller.
Vt is the tangential component of velocity and is positive in the clockwise direction.
Vn is the component of the velocity normel to the propeller disc and Is positive in the
astern direction.
z
in front of the results indicates that the values come from the extrapolated region.
e
indicates that it was not possible to perform the calculation for one of the two
perpendicular planes because the value was outside of the extrapolated region.
a) i 2 5 4
516171819
10111112
Point No. Speed Vin Pos.in the prop disc Measured pressures Calculated speeds r c p s h t vn/vm Vr/VIa v7vmalpha radius centre
port starb bottom top rn/sec deg.s me sin W mm W sin W mm W sii W 1 1.7759 560.0 60.0 110.5 1(8.9 108.8 102.5 116.5 e 5 0.190 2 1.7781 270.0 6o. o 112 . O 114.0 105.0 119.0 99.0 0.320 -0.1(8 0.150 5 1.7787 180.0 60.0 106.0 106.0 105.0 102.0 110.0 e-5 0.219 4 1.7805 195.0 60.0 105.0 106.0 101.5 100.0 110.0 e-5 0.195 5 1.7815 210.0 6o,o 106.1 109.0 99.8 106.1 105.2 0.220 -0.079 -0.078 6 1.7800 225.0 6o. o 108.0 111.0 96.5 115.0 94.0 0.545 -0.192 0.021 7 1.7819 240.0 60.0 109.5 112.0 95.5 121.0 87. 0 0.419 -0.206 0.105 8 1.7797 255.0 60.0 114.4 116.0 101.9 125.9 94.0 0.599 -0.163 0.159 9 1.7819 285.0 60.0 108.9 110.4 105.5 112.5 102.5 0.232 -0.058 0.126 10 1.7816 500.0 60.0 108.0 108.5 105.0 108.0 106.5 0.175 -0.055 0.045 11 1.7855 515.0 6o.o 108.9 108.6 105.0 106.5 108.5 0.236 -0.045 0.005 12 1.7835 550.0 6o.o 111.9 114.1 100.4 101.0 114.5 0.353 -0.144 0.022 13 1.7815 540.0 6o. o 111.0 116.0 97.0 96.0 117.0 0.552 -0.199 o.066 ib 1.7800 545.0 6o.o 110.1 116.0 96.2 95.0 118.0 z2 0.528 -0.218 0.090 15 1.7816 550.0 60.0 108.1 115.0 96.9 95.9 117.0 z) 0.279 -0.220 0.130 16 1.7822 555.0 6o. o 106.8 111.2 98.5 94.4 115.1 z2 0.254 -0.196 0.112 17 1.7757 360.0 60.0 167.8 170.0 162.5 159.0 175.0 z2 0.215 -0.160 0.091 18 1.7775 360.0 80.0 158.5 140.0 136.0 135.5 142.0 zS 0.146 -0.120 0.079 19 1.7771 357.5 80,0 158.5 143.2 152.2 151.5 145.0 z) 0.192 -0.143 0.145 20 1.7768 555.0 80.0 159.0 146.0 129.0 130.0 145.0 z5 0.240 -0.163 0.164 21 1.7784 350.0 80.0 141.0 151.5 123.6 128.0 155.5 zl 0.336 -0.177 0.177 22 1.7696 345.o 80.0 175.0 182.9 156.0 162.0 176.5 zl 0.528 -0.177 0.156 23 1.7721 530.0 80.0 175.0 182.0 156.0 167.9 170.1 0.400 -0.115 0.154 24 1.7743 515.0 80.0 173.0 179.0 157.9 174.0 164.2 0.359 -0.070 0.153 25 1.7746 500.0 80.0 173.0 179.8 157.5 181.0 158.0 0.549 -0.084 0.221 26 1.7743 285.0 80.0 176.0 181.0 157.0 192.0 150. 0 z2 0.444 -0.115 0.262 27 1.7762 270.0 80.0 180.5 181.5 158.0 206.0 159.5 z2 0.557 -0.158 0.293 28 1.7768 255.0 80.0 185.0 179.0 162.0 2114.0 155.0 0.662 -0.195 0.276 29 1.7781 240.0 80.0 187.0 177.6 164.0 215.1 132.0 0.703 -0.237 0.225 50 1.7809 225.0 80.0 185.5 178.0 165.0 215.0 134.0 0.671 -0.285 0.156 51 1.7787 210.0 80.0 178.5 179.0 160.2 198.0 156.5 0.504 -0.287 0.031 52 1.7805 195.0 80.0 169.1 169.6 165.0 167.9 168.0 0.225 -0.010 -0.052 35 1.7784 190.0 80.0 167.6 167.1 164.5 164.9 169.5 z2 0.182 0.083 -0.041 54 1.7816 560.0 110.0 105.2 104.8 104.0 104.7 105. 0 0.189 0.021 0.015 55 1.7746 360.0 110.0 105.0 105.6 104.9 105.1 '04.9
data are wrong.
56 1.8272 355.0 110.0 92.1 98.0 87.5 90.0 92.0 e-5 0.180 57 1.83 357.5 110.0 82.9 87.4 80.1 82.1 82.9 e-5 0.107 38 1.8292 550.0 110.0 85.2 99.5 69.2 80.9 84.9 e 2 0.260 39 1.7787 345.0 110.0 146.0 164.0 120.0 159.0 142.5 zl 0.405 -0.114 0.2614 40 1.7784 530.0 110.0 148.0 167.0 117.0 148.5 134.5 sI 0.454 -0.100 0.277 41 1.7787 515.0 110.0 150.0 167.0 117.0 163.0 122.0 0.490 -0.080 0.527 42 1.7790 500.0 110.0 165.0 120.0 179. 0 110.0 0.576 -0.092 0.357 43 1.7784 285.0 110.0 157.5 159.5 126.0 187.0 102.5 0.667 -0.115 0.338 54 1.7787 270.0 110.0 162.0 153.0 154.0 192.0 98.5 0.760 -0.155 0.3(8 5 1.7781 255.0 110.0 166.8 147.8 141.0 192.5 98.0 0.861 -0.158 0.261 46 1.7771 240.0 110.0 169.0 144.2 146.0 191.0 100.5 0.907 -0.181 0.218 47 1.7781 225.0 110.0 170.0 140.0 150.2 189.0 1(2.0 0.904 -0.196 0.184 48 1.7781 210.0 110.0 168.0 137. 0 151.0 188.5 101.2 0.879 -0.253 0.152 49 1.7771 195.0 110.0 1'40.O 140.0 135.0 147. 5 150.0 z2 0.264 -0.185 -0.010 50 1.7768 180.0 110.0 138.0 137.0 157.0 137.0 158.7 z2 0.157 0.054 0.000 51 1.7756 190.0 110.0 138.2 137.8 156.5 157.9 158.0 0.145 0.0(11 -0.021 52 1.7771 135.0 110.0 168.0 '45.5 155.0 185.0 105.0 0.911 -0.201 -0.156 HYDRa- 0g AERODYNAMISK Page LA0RA1URBJM
RERUL'18 of the Pr9Dr lUBE MEASUBEREN'rS
Lyngby
Denirrk
The test was run on the 22. 1. 1963.
The vater temperature was
The normal components are theoretically the same. Thus we have
three mutually perpendicular components which completely define
the flow at the point under consideration relative to the axes of
the pitot head.
For propeller calculations it is desirable to have the normal,
radial and tangential components of flow into the disc and these are
easily found if the position of the head in the propeller disc, and
the angle of the arm of the instrument to the vertical are known.
The actual computation of the three velocity components is
carried out at HyA by means of the GIER-digital computer. The source
program employed, together with the block diagram is given in
Appen-dix B.
A three dimensional wake survey was made for a single screw
slow running ship model by means of the five hole spherical pitot
tube and the measured values and results of the calculation are
given on typical standard data and result sheets (Pig. 7 and 8).
In the result sheet, data defining the position of measurement are
given together with the measured pressures and the calculated
non-dimensionalized velocities. Fig. 9 and 10 give a graphical
re-presentation of the results.
Accuracy
The diameters of the manometer tubes are not identical. In
addition air trapped in the water gives slight variations of
speci-fic gravity, which vary slowly with time, Other impurities are
also present in the system. Por these reasons the curve of
mano-meter readings against number of runs, Fig. 5, will vary slightly
with time and for different holes. Thus the equilibrium position
can vary for different tubes. These inaccuracies which are very
difficult to determine are small, less than i mm, and practically
constant during a run0
For a sufficiently high speed the measured water height will
be great in comparison with these inaccuracies0 This will always
be possible during calibration. From expression (2) it is clear
Fig. 9. The diagram of
the longitudinal
wake component.
V = the normal component n
of the speed.
V = the speed of the model.
m
Fig. lo. The diagram of the
velocity field in
the propeller plane.
12
-Looking to,ward
'so.
The wake survey was made
in a plane perpendicular
to the longitudinal axis
at a distance l.O6 LBP
forward of the A.P.
k
WL N 9_5\
\\\
OEiL 92u N 039L T lo Vn 09 0.9 0.7 6 0.5 a' 0.3 2 OElI
IWA
1(
r
w
360° 330° 3° 270e 240e 210°13
-This leads to difficulties when wake measurements are made on slow
rirn*ing models, i.e. less than tpDroximately 2 rn/sec., and these
difficulties are accentuated in che dead water regions at the top
and bottom of the disc0 A speed of .5 rn/sec. is about the lower
limit for accurate measurement with the instrument0
An accuracy of about i % can be expected in the region where
the water is not much disturbed by the model, but can be as low
as lO % or even more for slow running, full bodied models.
Concluding Remarks
Three-dimensional flow can be well defined from measurements
made with the IiyA five hole spherical pitot tube. When the
measure-ments are taken carefully, good results can be obtained.
Calibration curves were obtained at a rather high speed
and therefore no special scatter resulted. Inspection of the
calibration curves shows asymmetric tendencies which suggest
that the instrument could be slightly better, but better tools
and much more experience would be needed.
Inaccuracies can arise if the speed of the flow is low, as
is the case with slow running and/or full inodels.
The lower limit of acceptable accuracy is reached when speed
of the flow is .5 m/sec.
The inaccuracies can be smaller than 1 % if the speed of the
flow is high enough, but 10 % or more can be expected for the
lowest speeds.
Acknowl edgement
I would like at this stage to acknowledge my indebtedness
especially to Professor C.W. Prohaska, Director of the HyA Laboratory,
the Statens teknisk-videnskabelige Bond, and Mr. J. Strøm-Tejsen
for invaluable advice and encouragement.
Referenc es
Janes, C.E., "Instruments and Methods for Measuring the Flow of
Water Around Ships and Ship Modelst1,
David Taylor Model Basin, Report 487, (Mar. 1948).
Pien, P.C., "Five-Hole Spherical Pitot Tube",
pa-Po
- i - sin (- a
Thus it follows that
pa-pb
V2
n
-
14-APPENDIX
ADerivation of the ideal case
The following expression, which is derived from (i), applies
to point a0 Similar expressions are applicable to points b and c.
Y
l (sinh.b
-sin(.a)
-4
pc-pb - (sin (3-b - sin3.c)24
2pc-pa-pb (sin2(.asin2b-2.sin2f1-c) (7) - 4The velocity V of the flow is normal to the surface of the
sphere at the stagnation point, which is normally outside the plane
XY in Pig. a. The component of the velocity in this plane is
Via = V cos " , where is the angle between the XY plane
and
thevelocity. The following expressions give the relation between the
different angles:
cos (i-a = 2
sin i3a =
and similarly.2
sin (3.b.2
sin ,3c = Combining (7),i-cos2cos2 ((3h+)
1_cos2t. h(9), (io) and (ii) we
2pc-pa-pb (1-cos
-V'cos2
4 and (5), (9) and (10): pa-pb sin 2 2 2 4jÇV.cos
Dividing (13) by pa-pb 2pc -pa-pb Using (6), (io) pc-pb V2cos2 'cosX' . cos ((3-h -o<)
2 2 2
1-cos faa-a = 1-cos
(12), gives: 2ø). cos
(3
h-sin 2o( sin 2o( t an 1-cos 2o< and (ii):(cost3h
--4
15 -get 2/3h
2 (3-h cos2(,'3-h+))
16
-APPENDIX
BBlock Diagram and ALGOL Source Program of the
17
-Stock biaqram o the 5H SW Pitot Tube Calculotiov2
IS9RT I fte dota giVen t tile
( read aa mi odoq i prcted STOP f atta Or$ wrooq (priot8 te.N) area (t.priot.d
f
da&e re3d lOd lp -zfl i read Otsd. :r the roaqt C-ca(a) C4C.c .SSQ T '-WC.SG prats N,RøVi.TVm) dato Or. oc-sqrt(acM r oaq18
-ALGOL program Pitot Tube 5H SH begin
coent This is a program for the computation of velocity components in the propeller
disc1 from data obtained by means of the 5 Hole Spherical Head Pitot Tube. Different
symbols represent: Alpha and r are the angle and the radius in the propeller disc.
R is the arm of the instrument measured from the axis of symetry of the probe to the
point of rotation of the ans. Theta is the angle between the instrument arm plane and the vertical. Vm is the velocity of the model. The pressures measured at different
holes i.e. at the center1 port starboard1 top and bottom hole1 are represented by
c1ps1t and b respectively;
integer e1f1z1ab;
real R1Vm1ar1c1ps1t1b1ApiVy1Vz1ac1beta1N1S1CSQ1SG1V1CGRa1T1n;
array x[-l:1iJ;
procedure interpolation (x1c1h1yc); value x1c1h; integer c;real x1hyc;
begin
-ccnint here the data from the calibration curve are given; integer n; array y[0:23
if c = 1 then begin y[0] := 0.00; y[l] := 3.30; y[2] := 6.52; y[3] 9.79;
-
y[14] :=13.11; y(5] :=16.57; y[6} :=19.80; y[7] :=22.1+6;y[8] :=214.62; y[9] :r26.55; y[l0J:=28.12; y[ll]:=29.32;
y[12]:=30.25; y{l3}:=3l.; y[114]:=31.ß0; y[151:=32.147;
y[16]:=53.08; y{lT]:=33.60; y[18]:=314.0S; y[191:=314.149; y[20]:=314.84; y[21]:=35.l0; y[22]:=35.35; y[23]:=35.55; end;
if c = 2 then beT
yb]
:= 0.00; y(l] := 2.92; y[21 := 6.c4; y[3] := 9.52;-
y[14] :=13.11; y[5J :=16.146; y[6J :=19.59; y[71 :=22.37;y[8] :=214.76; y[9] :=26.71; y[l0]:=28.140; y{ll]:=29.92; y[12):=31.140; y[13]:=.52.65; y[114]:=33.78; y[l5]:=314.80; y[16]:=35.72; y[171:=36.62; y[16]:=37.148; y[191:=38.27; y[20]:=39.00; y[21):=39.69; y[22]:=140.32; y[23):=140.92; end;
if
c = 3thenbeT y[0] := 0.00; y[1] := 3.146; y[2] := 6.86; y[3] :=l0.32;-
-y[14] :=l14.42; y[5] :=18.57; y[6] :=21.96; y(7] :=214.98;
yt8] :=27.149; y[9] :=29.42; y(l0i:=30.90; y{llh=32.00;
y[l21:=52.145; y[13]:=33.60; y[114l:=314..30; y[15]:=35.; y[16]:=35.71; y[17]:=36.30; y[lB]:=36.814; y[191:=37.30; y[20]:=37.70; y[21]:=38.cTh y[221:=38.142; y[215]:=38.78;
y[0] := 0.00; y[l] 3.02; y[21 := 6.12; y[31 := 9.50;
y[141 :=l2.95; y[5] :=16.1i; y[6] :=18.95; y[7) :=21.1+9; y[8] :=23.73; y[9] :=25.70; y[l0]:=27.23; y[l1]:=28.148; y[1211:=29.53; y{13]:=30.51; y[114]:=31.140; y[151:=32.20; y[lG]:=32.96; y[l7]:=33.68; y[18]:=314.314; y(191:=35.00; y[20]:=35.61; y[21]:=156.22; y[22]:=36.83; y[23]:=37.141; end;
if o = 5 then beT y[0 := 7.80; y[l] :=11.32; y[2) :=13.91; y[3] ;=16.51;
-
-y[] :=19.9; y[5] :=21.69; y[6} :=214.27; y[7] :=26.6t;
y[8] :=28.714; y[9] :=30.50; y[10J:=3l.87; y{11]:=32.96; y[12]:=33.88; y(13]:=314.67; y[114]:=35.314; y{15]:=35.93;
y[16] :=36.148;
end;
if c = 6 then beI y[01 := 7.80; y[l] :=11.61; y[2] :=114.140; y[3] :=17.c9;
-y[14] :=19.67; y[5J :=22.10; y[6} :=214.146; y[7] :=26.69; y[8] :=28.80; y[9} :=30.70; y[10I:=32.31; y(llI:=33.149; y[12]:=314.11; y[13}:=314.514; y[1141:=314.79; y[15]=314.95; y[161:=.55.05; y[17]:=35.12; y[1B):=35.18;
end; end;
19
-if c = 7 then begin y[0] := 7.80; Y[U :=li.56; y[2] :=l4.O;
y[]
:=l7.0L;-
y[L] :=19.76; y[5 :=22.51; y[6] :=25.21; y[7] :=27.73;y[8] :=150.00; y[9] :3i.90; y[101:=33.51; y[il]:=3L.89;
y{12]:=%.il; y[13J:=37.20; y[iLJ:=38.20; y[15]:=39.12;
y[16]:=159.99; end;
if c = 8 then beT y[0] := 7.80; y[l] :=i1.56; yf2 :=1L.3l; y[] :=16.97;
-
y[] :=19.50; y[5 :=21.95; y[6) :=24.27; y[7] :=26.14;y[8] :=28.142; y(9] :=0.27; y[l0J:=l.B2; y[1l1:=32.97; y[12]:=33.71; y[1]:=L.20; y[lL]:=5L.L; y(l5]:=L.60; y{16]:='L.7l; y[i7]:=.80; y[l8]:=3t.8L;
end;
x:=abs(x);
-n := e-ntier (x/h); co-ne-nt -n is the subscript i-n the array; if n > 22 then begin e:=e+ab; C:=0; goto BB end;
if n>ll thc :=(x/h-n)x(y[n+i}-y[flJTy[n]The
:= ((x42)/(2xh42)_nxx/h+r42/2)x(y[n]-2xy[TT+y[n+2])
-(x/(2xh)-n/2)x(5xy[n] - 4xy[n+i] + y[n+2]) +
coinent the interpolation is made after Aitkens process of iteration by meàns of a second degree polynomial;
end interpolation;
comment model number 1model speed1 corresponding ship speed 1distance from the after
perpendicular shaft angle rake anle1 radius day month year and water
temperature corespond to A c1 s1 a1 N t1 R1 b1 S p and T respectively.
The se data are read and printed out in the heading of the re suits.;
tryktekst ('Ç<
H'IDRO- og AERODYNAMISK Page
LABORARIUM RESUL of the PITr '1JBE IASURi
Lyngby Denmark
POSITION of MEASURF2NT
Model no. ); tryk (Çnddd*1A); tryknl (26);tryktekst(Ç<Distance from the aft.pp.);
tryk (Çndd.dd a); tryktekst (< mm*); trykyr;
tryktekst ( 4odel speed); tryk(jndd.d&1*1c); tryktekst (i:< m/secl.); trykml(16);
tryktekst (<Sbaft a.ngle); tryk(ndd.d0*1N); tryktekst (<degrees); trykvr; tryktekst (<Corresponding ship speed ; tryk (Çnd.d*1s);tryktekst
(< knots Rake angle); tryk (Ç.nddd.dO*1t); tryktekst(<degrees
rykn1 (5L); tryktekst (Radius R'); tryk (4nddddd*1R); tryktekst (i:< mm);
trykvr;
tryktekst (The test vas run ori the*); tryk (ndd 1b); tryktekst (i:<. :);
tryk (nd*1s); tryktekst (<.); tryk (nddñ p);tryktekst(<.); trykvr;
tryktekst ($<The water temperature vas*); tryk ($nddd.dd*T); tryktekst (ç:< degrees centigrades. .); trykvr; trykvr;
tryktekst (j<
In the table bellow the actual measured and the computed results are given:);
T= isst;
c :=
isst; s := last; a := imst;N lest; t := lest; R := 1st; b := 1st;
,
comment The result sheet
heading hasnow been printed
and the calculation itself
begins;
again: n:=lst; if n=O then go to finally;
trykvr;
tryk(Çndddn);Vn: =lst; yk ( <ndd.
ddr/Vm);
a:=lst; tryk((nddd.dO-a); pi:=3.1J459265; a:=axpi/180;
for f:=-1 step 1. until do begin x[fJ:= lst;tryk(Çndddd.dx[f1); end;
T=i; z:=T;
cornxnt abz
and e are
auxiliary values directing the necessary jumps in the calculation;AA: A:=2xx[01-x[abI-x[ab+1];
if A < then
gin if abTEhen begin ab:=3;N:=O;e:=-5; gato AA end
begin C:=O;e:-e-5; goto nd; ¡;
A:=(x[ab+l]-x{TT/A; if A < O then abl;
-interpolation (Aab1/37 beta);Therpolation (betaab+L2.51V);
if beta > O then begin
if
ab < 5 then z:=z+l else z:=z2 end;if
A > O then beta:=-bet
ac:=x{OTtab];
if ac < Othen begin tryktekst(< data are wrong); gota again; end;
¡:=sqrt(a7J; s:cxsin(betaxpi/l80); C:cxcos(betpi7IU;
-B: if ab < then begin ab:=3; Vy:=S; N:xVm/2; gato kA. end
ase beginVT=S; N:=N+CxVm/2; end;
-inJ
C:=cos(a); n:=x[-lJ/TSQ:=sqrt(1-(Sxn),32);SG: =-S42xn+CxSQ; CG: =SxCxn+SxSQ; Ra:=VzxSG-VyxCG; T: =VzxCG+VyxSG;
if e=O then
gin ifjO then tryionl(L) else begin tryktekst(Ç< z); tryk(nz);
end; tryk(-nd.ddd*NRaxVm 'IxVn);
end else if e > ab then tryktekst Outside from the range)
1eTTelO then tryktekst(Ç<
data are wrong.*)
elseTnryktekst( et); tryk(-ne); tryk(Ç-nd.ddd2xÑTTend;
goto again;
rIlly:
tryktekst(<
20
-tryktekst (<
Vm is
the
model speedVr is the
radial
component of velocity and is positiveif
outwards from the centreof propeller.
Vt is the
tangential component of velocity and is positive in the clockwise direction.Vn is the component of the velocity normal to the propeller disc and is positive in the
astern direction.
z in front of the results indicates
that
the values come from the extrapolated region.e indicates
that
it was not possible to perform the calculation for one of the twoperpendicular
planes because the value was outside of the extrapolated region.);end Program 1 2 5 6 r 9 10
li[12
Point No. Speed Vm Pos.inthe
prop. disc
Measured pressures
Calculated speeds
r c p s b
t
Vn/Vm vr/vmvt/va
alpha
radius
centre port starb bottom topole//er
