HYDRODYNAMIC DRAG OF THREE-DIMENSIONAL BODIES 
BY MEANS OF A LASER DOPPLER WAKE SURVEY 
by 
JOHN RICHARD KNOBEL 
B. S. , Webb Institute of Naval Architecture 
(1973) 
SUBMITTED IN PARTIAL FULFILLMENT 
OF THE REQUIREMENTS FOR THE 
DEGREE OF 
MASTER OF SCIENCE IN NAVAL ARCHITECTURE AND MARINE ENGINEERING 
at the 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
(February, 1978) 
. · SiQnature redacted A 
Signature of Author •• •t:f'T .- .. -.............................. .,.... ............... . 
I\ Department of Ocean Engineering, February 1978 
· . . Signature redacted ~, 
Certified by ••••••••• Jr""•'• . -. . .-...................... ,..-.-. "Th~~;s· &!~~~~~~; . 
Signature redacted 
Accepted by •••••••••••.• ,.-~~":: ;:~~;;i:i~~~r:i~~;~~ ~~· ~~~d~~~~. ~~di~~ 
Archives 
HYDRODYNAMIC DRAG OF THREE-DIMENSIONAL BODIES
BY MEANS OF A LASER DOPPLER WAKE SURVEY
by
JOHN RICHARD KNOBEL
Submitted to the Department of Ocean Engineering
in February, 1978 in partial fulfillment of the requirements
for the Degree of
MASTER OF SCIENCE IN NAVAL ARCHITECTURE AND MARINE ENGINEERING
ABSTRACT
An investigation of the application of Laser Doppler Anemometry to
the determination of hydrodynamic drag by means of a wake survey is de-
scribed. The experimental study, which was limited to viscous drag, is
discussed.
The particular objects studied were a MIT Series of yacht keels,
with and without turbulence stimulation, at varying angles of attack, and
varying Reynolds number.
A discussion is given of the theory relating to the calculation of drag
from a wake velocity survey.
A series of guidelines are given for the use of the Laser Doppler
Anemometer System for future studies in the MIT Water Tunnel facility.
Graphs and tabulated results of the calculated drag coefficients versus
Reynolds number are given, along with a commentary on the particular results.
Name and Title of Thesis Supervisor: Justin E. Kerwin
Professor of Naval Architecture
ACKNOWLEDGE MENTS
The author wishes to thank his thesis advisor, Professor J. E. Kerwin
for his support and guidance in the writing of this thesis. He also wishes to
express his gratitude to Dean Lewis and Peter Minh for their assistance with
the experimental apparatus, and to Kathleen McGotty for assistance in pre-
paring the manuscript.
'1
CONTENTS
TITLE PAGE
ABSTRACT
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
1. INTRODUCTION
1.1 General
1.2 Wake Survey
1. 3 Laser Doppler Anemometer
1. 4 Limitations of the Study
2. THEORY
2.1 Wake Survey (General Case)
2.2 Wake Survey (Laser Doppler Anemometer)
2.3 Turbulence Stimulators
3. EQUIPMENT
3.1 Water Tunnel
3.2 Models
3.3 Laser Doppler Anemometer System
4. TESTING
4.1 Preliminary Tests
4.2 Production Testing
5. RESULTS
5.1 Keels In Smooth Condition
5.2 Results With Turbulence Stimulators
6. CONC I USIONS AND RE COMMENDATIONS
6.1 Experimental Conclusions
6.2 Recommendations
REFERENCES
APPENDIX I
APPENDIX II
APPENDIX III
APPENDIX IV
Data Reduction
Effect of a body on wake velocity in an ideal
fluid using a Rankine Ovoid to simulate the keel
Program Listing
Equipment List
1
2
3
4
5
5
6
6
8
10
12
14
14
17
22
26
29
35
47
48
51
52
55
57
64
5I. INTRODUCTION
1.1 General
Of considerable interest to all those engaged in the design of hydrodynamic
vehicles is their motive power requirement. Whether one is transporting a pay-
load from one point to another, or racing a sailboat, one is always concerned
with getting the most distance traveled in the shortest amount of time, with a
minimum expenditure of energy. This requires that the designer of any vehicle
to have a good understanding of the nature of the forces acting upon his craft, so
that he may in turn be able to predict their magnitudes and design the vessel
accordingly. In this particular case we are concerned only with the hydrodynamic
drag upon an ocean vessel, and the utilization of a fairly new tool, the Laser
Doppler Anemometer, to determine it.
1.2 Wake Survey
A well known and often used method of determining the drag of some
vehicle, or part of some craft, is by means of a wake survey. Any object
moving through a viscous fluid has in the region behind it an area where the flow
is disturbed to some extent. This might be due to flow separation, cavitation,
lift, or the existance of a boundary layer. A wake survey is a determination of
some property of the fluid in motion behind the body such that the drag of the
body in that condition of motion can be determined. In most cases this survey
takes the form of the measurement of the pressure at a series of discrete
points behind the form, and in this case the direct measurement of the velocity
behind the body.
61. 3 Laser Doppler Anemometer
The Laser Doppler Anemometer (LDA) is a device used to obtain the
velocity of a fluid flow without inserting any physical probes into the fluid,
which would in turn influence the flow patterns. The LDA device is actually
a system of electronic units and laser optics. This system is comprised of
the laser and its transmitting optics, the receiving optics, and the electronic
tracker with its signal processors. The laser generates a beam of coherent
light that is split into two beams, which cross at the point where you wish to
measure the fluid flow, with a known angle of intersection. Where the beams cross
they create interference bands, the size of which are determined by the wave-
length of the laser light and by the angle of intersection. With a perfectly clear
fluid these bands would not be visible, so the fluid is either seeded with small
particles, or the matter already suspended in the fluid makes them visible. Each
time a particle moving with the flow goes through these bands it in some way
scatters light. The receiving optics pick up these disturbances of the light
pattern, the frequency of which depends on the, velocity of the particle. The
particle velocity is assumed to be the same as that of the flow. The electronic
tracker is a device which determines the frequency of the particles crossing the
interference bands and generates a voltage which is linearly proportional to this
frequency, and thus proportional to the velocity of the flow.
1. 4 Limitations of this Study
The original intent upon the start of this work was to obtain the total drag
coefficient of a series of yacht keels, using the LDA, and to compare the values
found to those found by previous tests on the same models with a normal dyna-
7mometer. During the initial months of study it was found that this would involve
a number of points, and testing time so large as to be prohibitive. As this was
the first attempt at a drag claculation based on a wake survey at the MIT water
tunnel, the effort was directed towards finding out what measurements were
feasible with the LDA, and to develop techniques to use the instrument. After
becoming acquainted with the instrument, it was decided that with the present
equipment the only suitable investigation at this stage was a study of the viscous
drag to the keels, and to assume that in the mid-span they could be treated as
two-dimensional sections.
B
2. THEORY
2.1 Wake Survey(General Case)
The underlying theory behind all wake surveys is the conservation of
momentum. The survey consists of a set of measurements across the flow
downstream of the object being studied. Assuming that the flow is relatively
uniform before reaching the object, we hope to gain some information about the
hydrodynamic characteristics of the object by examining the disturbances created
by its passing. These can be either three-dimensional or two dimensional sur-
veys. In this case we are working with a small section in mid-span of the keel,
and are assuming it can be treated as two-dimensional. The drag hereafter
discussed is the drag per unit span, and does not include any tip or edge effects.
In this instance we are attempting to determine the viscous drag of a yacht keel
by calculating the difference in momentum between the non-viscous and viscous
flow pattern.
Using Fig. 2.1 as an illustration we shall assume a streamlined body
A 
_ _
with an axis of symmetry aligned with the flow. The flow into the control surface
surrounding the body is through face "A", and it has a momentum flux of;
I8 'P UJ2 dA,
9
The units of the momentum flux being ML/T 2 , mass per unit time multiplied
by its velocity. The additional assumptions needed in this model are those of
incompressible flow, and that the control surface boundaries are sufficiently
removed from the body so that the pressure has returned to its free-stream value.
The flow out of the control surface has a momentum flux of;
As the flow is symmetric about the bodies centerline, the only other possible
flow entering or leaving the control surface is through "S". By conservation of
mass this flow must have a magnitude of;
And, as it is leaving the control surface in the free-stream region, its momentum
must be;
These formulas deal only with the velocity and momentum components par-
allel to the centerline of the object. They are all readily expanded to include the
three-dimensional case. As "S" is far from the body it is assumed to have a
constant velocity, Us.
Summing the momentum flows into and out of the control volume we have a
term for the drag of the object;
DRAq =pS {(2- 2) - Us(UclJUj tA
10
In cases where the control surface "A" is very far upstream, and "S" is
far from the object the equation can be further simplified by equating Ua and Us.
This results in the form;
DRAQ =4Ub(Us--U) JkA
which is found in most texts, and even when all the assumptions required are
not met exactly, can still give fairly good results.
2.2 Wake Survey (Laser Doppler Anemometer)
In previous wake surveys of airfoil shapes, with the measurements of drag
as the intent, the pressure was measured at a series of points downstream of
the object. For practical reasons it was usually found difficult to measure the
flow sufficiently far from the object so that all the previously mentioned assump-
tions could not be met. This gave 'ise to a number of correction formulas,
notably those of Jones and Betz, as found in Schlichting. As pressure was the
quantity measured, these formulas list the drag coefficient of the object directly
in pressure terms, without ever finding the velocities.
The Laser Doppler Anemometer gives us the capability to measure the fluid
velocity at any point behind the object being studied without affecting the flow.
Thus the pressure terms are no longer needed to find the drag coefficient if one
can satisfy the assumptions of the pressure returning to free stream values.
If we were to attempt to measure the total drag of the keel being studied we
would again need corrections for the pressure field created by the foil if the
survey was done close behind the trailing edge. In this study we are concerned
only with the viscous drag of the keel. The reasons behind this specialization
ii
are discussed under "Preliminary Testing", Section 4.1. For zero angles of
attack the pressure drag for a streanlined body is small compared to the viscous
drag. For a fifteen percent thickness section the pressure drag is roughly three
percent of the viscous drag. If one is interested solely in the viscous drag one
can look at the momentum difference between the viscous and non-viscous case.
At this point we make the simplifying assumption that, at the point where we are
measuring the viscous flow behind the body, the observed flow outside the wake
is not significantly affected by the presence of the wake.
If this was so then the velocity difference in the wake, and the corresponding
difference in momentum between the experimental situation and the ideal case of
non-viscous flow would be the viscous drag. Rather than risk this possibly
unwarranted assumption, we will define the viscous drag found here to be that
found by the wake survey when comparing the momentum in the wake to the flow
that would have been in the wake if the actual pressure distribution (boundary layer
included) was present but no viscous forces. This is equivalent to adding the
momentum thickness to the outside of the foil and then treating it as a case of
ideal flow.
If we are considering a small region through the wake it appears reasonable
to assume that the pressure through the section remains r oughly constant. This
assumption is supported by Prandtl, as reported in Abbot and Von Doenhoff in the
simplification of the "Navier-Stokes Equations". The finding was that the pres-
sure was transmitted essentially unchanged through the boundary layer. Within
a wake the same considerations should still apply. A check of the velocity varia-
tions induced by a source-sink pair representing the foil is included in the Appen-
12
dix. The keel was modeled on a Rankine Ovoid, a "worst case", and the velocity
differences were calculated. They were less than one percent of the flow velocity
and are believed to be low enough to ignore in the present study. With these
assumptions we can consider a non-viscous wake profile to be similar to "A" in
the previous example, and our viscous profile as boundary "B". As the two
profiles are the same outside the wake region the integral need only be evaluated
over the wake. The assumption is still made that the flow leaving the boundary
transversely, through "S", leaves at axial velocity Us.
2.3 Turbulence Stimulation
As previously discussed, the drag of an object in a mixed laminar and
turbulent flow is dependent on the extent of the laminar region. The NACA 66
series of foil shapes tries to maximize this region by delaying the location of
the increase in pressure on the surface of the foil. As the extent is also dependent
upon the Reynolds number and the surface roughness these items have to be taken
into account when expanding model tests to full size. Not only must the Reynolds
number of the model match that of the full size section, but the regions of laminar
and turbulent flow should be as close to identical as possible.
It is widely believed that yacht keels operate in an almost full turbulent
fashion on any but extremely small craft. In order to simulate this in testing
facilities for keels and hull shapes, it is common practice to install some kind
of turbulence stimulator to the forward edge of the body being tested. It is hoped
that this device, which could be a wire, or a row of pins, or sand grains, would
cause turbulent flow to exist over the same region on the model as is experienced
by the full size shape. Any of the devices mentioned do this, but also add their
13
own drag forces. They may be located in such a fashion that the laminar flow ahead
of them will cancel the effect of the increased drag, or the estimated added drag
may be subtracted from the tesL results. The precision of the LDA should make
it possible to see what effect these devices have upon the drag.
6
14
3. EQUIPMENT
3.1 Water Tunnel
The testing was done in the MIT Department of Ocean Engineering Water
Tunnel. This is a recirculating flow, variable pressure, variable speed twinel
with a constant cross section through the test area. With no object in the test
section the flow through that section is usually quite uniform. The models of the
yacht keels were mounted on a dynamometer that could vary their angle of attack.
The test section incorporates a splitter plate to minimize the effects of the upper
boundary layer of the flow on the flow at the model. The test section arrangement
is shown in Fig. 3.1.
3.2 Models
The models tested were an MIT series of yacht keels. These were devel-
oped under sponsorship of SNAME. Previous testing with these keels was done
at the MIT water tunnel and is reported in "Water Tunnel Tests of a Series of
Yacht Keels with Varying Reynolds Numbers", by Kerwin and Lewis. This is
an unpublished paper for SNAME research panel H-13. The illustrations of the
test section and the keels are taken from that report. Their platform, which was
the same for all of the models tested for this work, is shown in Fig. 3.2.
The models were mounted in the same fashion as the previous study, so
that the results would be directly comparable. All three keel models were
tested in the smooth condition, and model one was tested again with the addition
of turbulence stimulators.
The turbulence stimulators consisted of number 20 carborundum grit.
15
SPLITTER PLATE 
\KE
FLOW
BOTTOM WALL
SIDE VIEW OF TEST SECTION
KEEL
CROSS SECTION 17-3/4"
20"
Test SectionFig. 3. 1
8.0" Root Chord-1
40
Sweep
TC~
Fig. 3.2 Keel Planform
Models 1,2 & 3
Model 1 632-015
Model 2 662-015
Model 3 63-010
Fig. 3.3 Keel Model Sections
8.0" Tip Chord-"I
I
I
These particles were epoxied to the surface of the keel. The keel was first
tested with the addition of one row of the stimulators ten millimeters behind the
leading edge, and then later with another row added five millimeters further
back. In both cases the spacing of the individual grains was roughly three grains
per centimeter. It was originally planned to add a third row and test the model
again, but this was not done due to a lack of available time in the water tunnel.
The keel cross sections are shown in Fig. 3. 3, and additional information
on their development can be found in the report by Kerwin and Lewis.
3. 3 Laser Doppler Anemometer System
The Laser Doppler Anemometer System is composed of two optics groups
with their individual power supplies, and an electronics gaoup which converts
their output to a voltage.
A list of the individual items of equipment is included in the appendix.
Referring to the pictorial and wiring diagram, Fig. 3.4, the following is a brief
descriptinn of their set-up and operation.
The laser and transmitting optics are mounted together on a single baseplate.
The transmitting optics take the single beam of light produced by the laser and
split it into two parallel beams. In the set-up used for this test the beams were
then passed through a frequency shifter. This device consists of a set of prisms
and a Bragg cell. The Bragg cell operates on a piezeo-electric principle, and
shifts the frequency of one of the beams a selected amount. The selection of what
frequency to be used is determined by the speed of the flow being measured, the
particle size, and the optics being used. The frequency shift used throughout this
Receiving
Optics --
Photo-
detector
Power Supply
Keel
Frequenc
Shifter
Bragg
ceL
y
Laser
iLL
Laser
Power Supply
Counter
(Impeller)
Tracker &
Signal
Conditioner
Voltage to
Frequency
Converter
C Counter
(Velocity)
Fig. 3. 4 Pictorial/Wiring Diagram
IC
19
work was one megahertz. There are adjustments on the transmitting optics to
adjust the relative brightness of the two beams, and the alignment of the shifted
and the unshifted beam. After the frequency shifter the beams pass through a
lens of known focal length. When property adjusted, the shifted and unshifted
beams should intersect at the point where one wishes to measure the fluid velocity,
with a known angle of intersection. The velocity will be measured normal to the
line that bisects their crossing point.
In the test set-up at the MIT water tunnel the beams travel through the plexi-
glass wall of the test section before intersecting in the flow. Any imperfections
in the walls will deflect and/or obscure the laser beams. Any deflection from
their intended path will induce an error in the ultimate velocity reading, while if
one of the beams is partially blocked it will decrease the usable signal, which
could cause the electronics to track noise rather than the proper signal. The
plexi-glass walls used were the best of those presently available at MIT but are
still considered to be only marginally adequate. The imperfections were essen-
tially random and no attempt was made to correct the data for these effects.
The receiving optics are mounted on a separate baseplate on the opposite
side of the test chamber. They were adjusted to be parallel to the transmitting
optics. As the receiving lens does not send any light through the walls, but rather
collec ts the light scattered by the intersection of the two laser beams, the wall
imperfections are only important as far as blockage is concerned. The optics
focus the light upon a photo-detector. The induced voltages in the detector are
then transmitted to the electronics.
20
The electronics are based around a tracker, or frequency following device.
This unit senses the frequency of the signal from the photo-detector, which is
that of the particles going through the beam intersection. The tracker generates
a voltage directly proportional to that frequency, as the frequency is directly
proportional to the velocity the voltage is proportional to the velocity as well.
The electronics package also contains a signal conditioner to eliminate noise
effects from the output and to bring the output voltage into a region where the re-
cording maters were most sensitive.
The values of the adjustments on the signal conditioner and the tracker for
all the tests were; one volt output per megahertz of the input signal, a five kilo-
hertz low pass filter on the output, and a one volt suppression to eliminate the
voltage induced by the frequency shifting. The low pass filter was used to elimi-
nate transients and noise from the output. As we are averaging the flow over ten
seconds it seems reasonable that any fluctuations in the flow greater than five
kilohertz could be ignored.
The output from the tracker is in discrete voltage levels. The tracker will
sense the input frequency, match that with its output voltage, and electronically
verify the result. This results in the output of a square-wave type, centered
around the voltage which would represent the average velocity. Viewing this
output on an oscilloscope was found to be very useful in adjusting the various opti-
cal stages and the electronics.
In order to average the frequency over ten seconds the output of the tracker
was then fed to a voltage to frequency converter, whose output was in turn fed to
a frequency counter reading over ten seconds. It is felt that this is not the best
21
way to get numerical results, as each instrument added to the system introduces
some error, but it was the only system available.
22
4. TESTING
4.1 Preliminary Tests
Whenever a new instrument is developed and put into use in the laboratory,
there is inevitably a period of familiarization necessary before the tester can
have confidence and expertise in its use. The LDA system is a new concept in
flow measurement, and this particular study was the first of its type at the MIT
water tunnel. Throughout the summer months of 1977 preliminary tests and wake
surveys were done using the LDA and keel number one.
The initial experiments were not intended to yield numerical values, but
to explore what regions of investigation would be open to this device. The
original intent was to develop a method to calculate the total drag of a three-
dimensional object, in this case a yacht keel, by means of a wake survey. This
would be the sum of the viscous drag, tip and root drag, induced drag, and form
drag. In order to accomplish this it would be necessary to have a wake survey
that covered the entire test cross section behind the model. To account for
effects caused by the wall boundary layers, and keel and wall interactions, it was
thought that it would probably be necessary to do at least a partial survey in front
of the keel model. From the initial work, and the time it took to make a single
traverse and record the data, it was obvious that there would not be enough time
to do this type of survey for all the models in the MIT keel series. Within the
allowable time the available options were to do a complete study of one keel at a
limited number of Reynolds numbers, or to investigate one component of the
flow around a greater number of models at different speeds. A factor that
influenced the decision was the results of an earlier study done on these models
23
at the water tunnel, with the standard dynamometer. Those results were drag
coefficients substantially greater than those expected. It was decided to investigate
the viscous drag component of a number of models at different Reynolds numbers
to compare with this earlier data. It was thought that a survey of one keel, even
if much more extensive, would not be as useful.
Once it was decided that the wake surveys were to be limited to the viscous
effects the assumption was made that near the midspan of the section the flow
was not seriously effected by tip and root effects, and could therefore be
treated as two-dimensional. As the models are of keels with a forty degree
sweep, this assumption may not be made lightly. It was thought that limiting
the angle of attack to small values would necessarily follow this assumption.
A considerable portion of the early tests were devoted to checking this assumption.
With the admittedly crude data reduction method in use at that time it was decided
that above six centimeters from the keel tip that the two-dimensional assumption
was not unreasonable.
Items of particular inportance that came to light during this trial period
were the following:
The output voltage readings of the instruments were to some extent depen-
dent upon the gain level set on the tracker. That is, turning the gain above a
certain level in order to obtain higher data rates, resulted in picking up large
amounts of background noise. This noise was for the most part thought to be stray
light from the laser and which showed up on the oscilloscope as zero velocity
readings. Thus with the gain turned up one could run the water tunnel at any
velocity and still have a reading of zero velocity. The means used to avoid this
24
condition were to observe the output of the tracker on the oscilloscope. A gain
setting that was too high would show the output jumping towards zero voltage.
A plot was made of data rate versus output at a constant water tunnel speed. The.
point where this deviated from a straight line corresponds closely with the abnor-
malities seen on the oscilloscope.
It was found that with the model in place the flow across the tunnel was
fairly uniform, except for the wake of the model and the tunnel wall boundary
layers. The most significant departures from this being at high velocities with
large angles of attack. For this reason it was decided to do a survey of the keel
wake that only covered a transverse distance large enough to determine the "free
stream" velocity on both sides of the wake. This allowed the use of a transmitting
optics lens of a smaller focal length, and reduced the required number of data
points on each run.
In a survey of this type time is one of the limiting factors. The periods
when the water tunnel is available are limited, and a survey with many discrete
point data readings requires large amounts of time. As the testing was to be done
over a range of Reynolds numbers, the tunnel speed as well as the transverse
position of the laser had to be varied. In the preliminary period the method of
changing velocity at a constant location, and that of holding the speed constant
while moving the laser from point to point were tried. It was found that working
at a constant speed was slightly faster, as well as wliminating some concerns about
the transient speed response of the water tunnel.
The facility at present has no method of changing the streamwise location of
the laser, which is the location of the survey. For these tests with sept back
keels, the tip of the keel was used as a reference point. The distance behind the
keel was varied by changing the height of the laser.
As the velocity in the tunnel was not truly constant over the time it took to
make a traverse, it was averaged over ten second intervals. The drive motor
RPM was averaged and recorded over those same ten seconds. By varying the
speed of the drive impeller, and reading the laser voltage while at a "free stream"
position it was found that for practical purposes the flow in the tunnel was linear
with the impeller RPM. This was used to normalize the recorded data in the laser
wake surveys.
26
4.2 Production Testing
Keels number one, two and three were tested in sequence, the primary
nurveys being done at eight centimeters above the tip of the trailing edge. The
laser was aligned with the tip of the keel while at zero angle of attack, and then
raised to whatever height was being investigated. This resulted is the survey
being slightly over six point seven centimeters behind the trailing edge of the
keel at zero angle of attack. This distance increased with increasing angle of
attack as the laser was left in a constant position and the keel rotated away from
perpendicular to its axis. This should have no effects on the results as they
should be independent of streamwise position.
Keel number one was later retested with the addition of turbulence stimula-
tors as previously described. The tracker experienced some difficulty with the
highly turbulent flows. It was much harder to obtain a high data rate, and losing
track of the data was much more frequent than in the other tests. Possible reasons
for this are the mixture of fluid flow velocities and directions and unsteady flow.
If the flow contains fluid moving at different velocities, or at the same velocity
but in different directions, it is possible that the bandwidth of the tracking elec-
tronics is being exceded. If this were the cause then a switch to manual tracking
might help. If this effect is caused by the unsteady nature of the flow, the solution
would be to average the flow over a greater period of time, or to complete the
entire traverse much more quickly. Switching the tracking unit to manual in this
instance would result in a bias of the data obtained. As these problems were
encountered at the very end of the test period it was not possible to determine
which of these, or some other influence was causing the misbehavior. A switch
to a different frequency shift, in view of the high frequency components of the
turbulent flow, might be of some help but it was not found possible to try this
in the test period.
The procedure used to take one data "point", which consists of the voltage,
impeller, RPM, and position was the following:
a) Check the RPM as shown on the counter and adjust to within one RPM
b) Posit ion the Laser and Tracking optics on the desired point.
c) Check the output of the tracked and the input signal to the tracker on the
oscilloscope. In this way one can tell if the optics are focused, if one is receiving
a noisy signal, or if one is tracking one of the noise signals.
d) Reset the voltage and RPM counters at the same time. As these were
both ten-second counters, one can then read them simultaneously.
e) Plot the point thus obtained to check if it agreed with previous points.
In some cases the tracker would lose the signal part way through a ten-second
cycle. In this case the error would be small enough that it could slip through
unnoticed if the data were not graphed.
Most of the testing consisted of measurements wither inside or adjacent
to the wake. The spacing between data points across the wake was usually one
millimeter, and outside the wake from one to four millimeters. In most cases
the data was taken in two traverses, one for odd numbered data points, and another
traverse in the opposite direction taking, 'the even data points. In this fashion it
was hoped that any irregularities caused by temporary misalignments or malad-
justments would be quite visible on the graph.
27
28
The LDA system is a device for precise, and literally pinpoint velocity
measurements. One of the chief drawbacks with the present installation for this
type of study is the time required to make these measurements. With everything
running smoothly the typical time for a traverse would depend on the number of
data points being taken. The time required for each data point is approximately
one minute. If one is receiving a weak signal from the optics, which could be
caused by any number of factors; misadjustments in the equipment or flow
conditions, one could expect to spend two minutes or more per data point until
this condition is corrected.
The data was reduced by the use of a computer program, written in FORTRAN,
for use on the MIT 370/168. The program is based on fitting a polynomial through
the wake velocity profile and calculating the difference in momentum between the
viscous and non-viscous flow. A program listing and further explanation are
found in the Appendix.
29
5. RESULTS
5.1 Keels in Smooth Condition
As can be seen from the graphs of drag coefficient versus Reynold's
number ( Fig. 5.1, 5.2, 5. 3 ) all the keels exhibit generally decreasing values
of drag coefficient for increasing Rn at low angles of attack. The two keels with
fifteen percent thickness exhibit this tendency to a greater extent than the keel
with a ten percent of chord thickness. (Keel no. 3) At six degrees angle of
attack one starts to see a rise in the drag coefficient at increasing Reynold's
number, this tendency also exhibited by Keels nos. 1 and 2 greater than number
three.
A quick look at the ITTC correlation line, of the ATTC friction line will
indicate that these keels were not operating in a fully turbulent manner. As the
drag coefficients here are based on the chord of the keel, they would have to be
divided by two to get an equivalent flat plate friction coefficient. This would put
6
typical values at about 0. 003 at a Rn of 1. 0 x 10 . The Blasius solution for the
drag coefficient of a flat plate in laminar flow is:
1
Cd = 1. 328/(Rn)2
The corresponding drag coefficient for a flat plate at Rn =10 6 is 0. 0013,
so one is clearly operating in a region of partial laminar flow. In this case the
drag coefficient is directly dependent on the extent of the laminar boundary layer.
The transition point from laminar to turbulent flow will depend upon the pressure
distribution in the flow. The laminar region can not exist to any extent in areas
of increasing pressure. Comparing the results to those of Squire & Young, again
3 0
in Schlichting, at Rn of 106 and using a ten percent thickness ratio, the drag
coefficient is seen to vary from .012 at no laminar flow, to 0.008 at a transition
point of 0.4 chord. The values obtained in this study were a good bit lower than
these predictions, which would seem to indicate a greater region of laminar flow.
The results agree with those published in Abbot & Von Doenhoff for keels
numbers one and two within the limits of experimental error. While their figures
are at a considerably higher Reynolds number, the drag coefficients are substan-
tially the same. Abbot & Von Doenhoff do not list figures for keel number three,
the 632-010 section, but the data agree fairly well with that on the 632-009 section.
It should be mentioned that the data in Abbot & Von Doenhoff was obtained by wake
surveys of the pressure type.
A comparison with the results Kerwin & Lewis shows that the values ob-
tained in this work were approximately one-half of the drag coefficients obtained
with the dynamometer. The data follows the same trends as observed in their
work, but a decrease in drag coefficient from low values to about 1. 0 x 10 6 and
then a leveling off with increasing Reynolds numbers.
C"j
0D
Keel No. 1
0.0 deg. angle of attack
4.0 cm above tip
6.0 deg. angle of attack
8.0 cm above tip
4.0 deg. angle of attack
8.0 cm above tip
j I I I 1 . I
- 2.0 deg. angle of attack
8.0 cm above tip
* 0.0 deg. -angle of attack
8.0 cm above tip
1 I I I I I
.2 .4 .6 . .0 12 1.4 .6
Reynold's No. (x. 10 -6) Fig. 5.1
81
C4
C%4
C%4
C'4
Keel No. 2
6.0 deg angle of attack
8.0 cm above tip
4CM'
_____ 
I I I I I I I I I A I ti
4.0 deg. angle of attack
8.0 cm above tip
I . _ . I .. ... I I 
-.-- I -
2.0. deg. angle of attack
8.0 cm bove tip
1 1 I I I I I I .
0.0 deg. angle of attack
8.0 cm above tip
.2 .4 .6 .8 /0 1.2 1.4 /.6
Reynold's No. (x 10-6) Fig. 5.2
0
0
Cl
0
r-I
0
- -
4ci
4-1
Q)
0
c-)
to
p
cl
Cl
0
0
0
33
Keel No. 3
-- 4
6.0 deg. angle of attack:
8.0 cm above tip
r-4
0-
'-A.. I I
4.0 deg. angle of attack
0 8.0cm above tip
0o
44
0
U CI4 2.0 deg.,angle of att ack
-8- 0' cm above tipC)
0.0 deg. angle of attack
8.0 cm above tip
_4 _ _ .L '8 1.0-1.
.2 .4 .6 .8 /.0 1.2 /.4 1.6
Fig. 5.3Reynold's No. (x 10-6)
Comparison of Keels 1,2 & 3
c
Lm
0 C)
Lfl
r-4
0
0
04
0
0)
cv,)
x KEEL NO..I
02-
'I j6 3
6.0 Deg. Angle
8.0 cm above
tip
A .&. -A - .. a i & e a A a I a .
4.0 Deg. Angle
8.0 cm above
tip
-A .i . 1 I I I -
2.0 Deg. Angle
-8.0 cm above
tip.
... I I L 2 I . I-
0.0 Deg. Angle
8.0 cm above
ATTC Line 7;?
.2 .4 .6 .8 /.0 1.2 .4 /.6
Reynold's No. . (x 10 -6)
34
04
-0
04
0
C-,
4-)
mr4
4-4
a)
0
0
0
0
Fig. 5. 4
:i35
5.2 Results with Turbulence Stimulators
The graphs, Fig. 5. 5, presents the results of the testing of keel no. I
with the addition of the carborundum turbulence stimulators. There is consid-
erably more scatter in this data than in the data for the keels in the smooth con-
dition. It is believed this is a result of the electronics of the system having
greater difficulty in tracking a turbulent flow. As the wakes for smooth keels
are also turbulent it is conjectured that this might be unsteady rather than turbu-
lent effects. As this testing was not completed until shortly before this writing
there was no time to investigate the source of these effects.
Plotted on the zero degree angle of attack graph is the ATTC friction line,
as developed by Schoenherr. The general trend of the data appears to be a
gradual decrease in drag coefficient with Reynolds number, but with less slope
than predicted by the ATTC line. The difference in drag coefficient produced
by the addition of the second row of turbulence stimulators is slight. There is
not enough data to develop a good estimate of the drag of a row of stimulators.
The results do tend to indicate that the turbulence stimulators add a roughly
constant increase in drag coefficient when compared to the keel in smooth
condition. An interesting effect is that the drag coefficient does not appear to be
influenced a great deal by the angle of attack when the turbulence stimulators are
added. This is thought to be a scale effect, which unfortuneately was not inves-
tigated further, as it might have some significance in other model tests.
36
Keel No. 1, The Effect of Turbulence Stimulators
.. *. -'0.D77D V7Aii;j
- -- -__ - -- 4 . -c- -t -ta
-- ---.------ - -- *- - -- *-_-- -- 4
-.. 
... ... .. - one--row :
-two-rows
-7.- - -.- . -.-. - .- - 1 - ---- -"- - - -' -
.-- -
6.0 Deg. Angle of ett-aek
o : .n-7r.I- -- a O - - -
7-
__ .. p i . .t . a a I U
4 0 Deg. Angle of attack
8-czrabovEt:
C)C
7 --... .-. --
0+ - -- egQ. Angle of attack
o -... r - - -t-- - ---z -r
---1- -'C
.2 . 6 . ... .IA-- - -,1.- 16-- --
Reynol d' N. C- 10-6)-Fi
- --- - - - -- - " -T . .--- k e
.2 .4 .6 .8 /.0 1.2 1.4 /.6
Reynold's No. (x 10 -6) Fig. 5.5
Table 5.1 Keel No. 1,- Smooth condition
Test Date Water Temp. RPM Angle Hgt. V avg. Rn Cd Cd(S)
(OF) (deg) (cm) (cm/sec) (x1-5)
1_1 _ __ I IIIIII
"
11-1 2-77
"f
"
"'
"1
11-1 3-77
"1
11-11-77
" -
?? .0
"U
"'
"1
"I
78.5
IV
21".
"'
77.0
"3
"l
78.0
50
100
150
200
300
50
100
150
200
300.
50
100
150
200
300
50
4
8
.0
so
"
"
0.0
I"
"'
"U
"U
0.0
"U
"'
"'
"
2.0o
'3
"'
'3
"U
4.0
a - a &
101.2-
203.4
298.3
4oo.8
603.3
99.5
205.7
306.6
407.4
61o.6
101.5
205-7
306.7
407.7
609.8
101,7
2.29
4.61
6.76
9.08
13.7
2.30
4.80
7.08
9.41
14.'
2.30
4.66
6.95
9.24
13.8
2.33
0.01381
0.00927
o.oo605
0.00695
0.00621
0.00636
o.o0684
0.00604
0. 00568
0.00514
0.00839
0. 00715
0.00639
0.00688
0.00579
0.01136-
0.01440
0.0939
0.00653
0.00701
0. 00627
0.00664
o.o0694
0.00679
0.00571
0.00516
o. o0867
O.00720
o.oo646
0.00691
0.00587
0.01178
"U
8.o
'3
"f
"9
"s
8.o
"'
"'
"U
11-13-77
Table 5.1 (continued)
Test Date Water Temp. RPM Angle Hgt. V avg. JCdCd(S)
(OF) (deg) (cm) (cm/sec) ( x10-5)
11-13-77
"t
4.69
7.02
9.31
14.0
2.26
4.64
6.89
9.21
13.8
9.00678
0.00621
0.00581
0.00453
0.00911
0.00808
o.oo639
0.00652
0.00869
0.00684
o.oo644
0.00595
0.00457
0.0101
0.00813
o.o0661
0.00662
0.00844
I
I --
cc
"f
11
"'
-14-77
"3
78.0
"3
3'
3'
77.0
'S
"'
"I
'3
100
150
200
300
50
100
150
200
300
8.o
'3
"I
"I
8.0
",
"'
4.0
"
"
",
6.0
to
is",
"I
3'
'3
"'
204.7-
305.8
405.5
608.2
99.5
204.8
304.0
406.0
608.1"
S .3 .& '.
Table 5.2 Keel No.
Test Date Water Temp. RPM Angle Hgt. V avg. [Cd [ Cd(S)
(_F) (deg) (cm) (cm/sec) ( x 10-5)
11-15-77
",
"f
",
"1
11-17-77
"o
of
"U
"U
11-18-77
"9
"'
"'
"
11-19-77
________________I I .1 1~ 'S
77o
it
",
"'
"'
77.0
"'
"'
"'
"'
77.0
"9
"'
"1
"'
77.0
50
100
150
200
300
50
100
150
200
300
50
100
150
200
300
50
2
000
2.0
of
of
it
It
4.0
-of
"t
"90
8.o
"U
"U
"'
"U
8.0
1
of
8.0
8.0
98.5-
204. 3
304.1
407.5
608.4
102.1
203.0
303.4
404.4
604.7
10089
203,. 1
303.5
404.1
605.4
100.5
2.23
4.63
6.90
9.24
13.8
2.31
4.60
6.88
9.17.
13.7
2.29
4.6o
6.88
9,.16
13.7
2.28
0- 00805
0-00762
o. 00437
O.00554
0.o0489
0. 01265
0. 00759
0. 00726
o.oo6os
0. 00457
0901020
0. 00703
0. 00708
0.00546
0.00458
0.00902
0.01086
0.00871
o. 00652
09 00563
0.00492
0 901305
0.00838
0. 00728
0 .0060 2
0.00482
0.01163
0 .00789
0.00699
0.00566
o.oo469
0*00978
I,
v
Table 5.2 (continued)
Test Date Water Temp. RPM Angle Hgt. V avg. J1 CdI Cs(S)
(OF) (deg) (cm) (cm/sec) x10-5
_______ ( x 1o0 __ _ _ _ _ _ _ _
11-19-77
11-19-77
11-19-77
-I
77 .0
Is
77 0
77.0
100
150
200
300
50
50
6.0
"'
"'
6.0
4.0
8.0
",
'3
4.o
4.0
201.5-
300.9
402.3
6o1.6
100.4
100.4
4.57
6.82
9.12
13.6
2.28
2.28
0.00705
O.00727
0.00744
O.00709
0.01226
0.01659
0 .00758
0.00818
0.00788
0.00725
0.-01255
0.01728
.1______________ 
___________ L I
Test Date Water Temp. RPM Angle Hgt. V avg. Rn Cd Cd(S)
(OF) (deg) (cm) (cm/sec) ( x 10-5)
of
"U
"'
11-20-77
of
to"'
"
11,-20-77
"t
I
I
77*.0
"'
"U
"'
"'
78.0
1U
U,
"U
"U
78.0
19
79.0
50
100
150
200
300.
50.
100
150
200
300
50
100
150
200
300
50-
0.0
it
"U
"'
"
2.O
go
of
I"
of
4.0
"1
"U
"'
6.0
8.0
,
I
"'
8.o
11
If
t'
it
8,o
"'
"'
"U
"'
8.0
102.7
205.9
307.4
408.5
613.7
102.7
206.4
307.3
408.4
611.9
102.9
206.5
306.9
407.8
610.3
101.9
11
2.33
4.67
6.97
9.26
13.9.
2.36
4.74
7.05
9.38.
14.0
2.36
4.74
7.05
9.36
14.o
2.37
-20-77
"
0.00671
0.00525
0.00510
O.00430
o. oo431
0.00595
0.00538
0 900521
0.00470
0.00447
0.00623
o.00605
0.00581
0. 00535
0900514
o.00634
0.00686
0.00542
0.00517
o.oo464
0.00436
0.00636
0.00560
O. 0526
0.00478
0.00451
0.00664
o.o0610
0.00584
0.00541
o o0520
0.00704
"0
"U
U'
11-20-77
S & 1.
I
pIll -1
I
I
I " m HTM
Table 5.3 (continued)
Test Date Water Temp. RPM Angle Hgt. V avg. .[ Cd Cd(S
(OF) (deg) (cm) (cm/sec) x10-5)I1C-(S
11-20-77
it
i,"'
799.0
11
",
"S
100
150
200'
300
6.0
"'
'3
"I
8.0
"'
"U
"I
205.1
305.9
407.3
608.7
4.77
7.11
9.47
14.1
0.00744
0.00751
0 .00732
o.00665
Sm~9 t b U .~ ~ ft
0.00756
0.00758
0.00741
0.00675
Table 5.4 Keel No. 1 With one row of turbulence stimulators
Test Date Water Temp. RPM Angle Hgt. V avg. Rn Cd Cd(S)
(OF) [I _ I(deg) I(cm) [I(cm/sec) (x10-5) [ [
12-17-77
I
"3
12-1 9-77
3,
"f
i"
"3
12-20-77
1"
'3
12-21-77
79.0
so
to
'3
if
79.0
"3
"
'3
"3
79.5-
9.
80.
50
100
150
200
300
50
100
150
200
300
50
100
150
200
300
50
0.0
31
",
"I
1
2.0
"'
"'
'3
"
4.0
4.0
"l
't
.0
"'
"'
'3
3f
.0
"
"'
"3
"3
8.0
98.1
202.8
298.9
400.2
.598.9
102.4
20-2.8
306.7
405.8
607.3
102.7
203.7
303.9
404.7
606.0-
99.7
2.28
4.71
6.95
9.30
13.9
2.38
4.71
7.13
9.43
14.1
2.40
4o75
7.11
9.46
14.2
2.35
0.01818
0.01737
0.01508
0.01748
0.01412
0.01415
0. 01271
0 s01358
0.01253
0. 01171
0. 01598
0.01404
0. 01330
0.01275
0.01174
0.01460
I M ~ I ~ ~'. A
0.01859
0.01747
0. 01557
0.01781
0.01439
0.01442
0. 01311
0.01354
0.01259
0.01176
0. 01604
0.01414
0. 01347
0.01289
0.01185
0.01483
"-'" I
8
8
Table 5.4 (continued)
Test Date Water Temp. RPM Angle Hgt. V avg. Cd Cd(S)
(OF) (deg) (Cm) (cm/sec) x 10-5)
12-21-77
"e
12-21-77
I
i,
"1
"
80.0
"f
i ,
80.5
",
"'
100
150
200
300
50
100
150
200
300
4.0
"U
El
EU
6.0
U,
U,
EU
"U
8.0
EU
",
U,
8.o
of
tU"t
"t
201.3
300.5
402.7
604.6
100.7
202e5
303.6
402.6
602.3
4.74
7.07
9.47
14.2
2.37
4.76
7.19
9.53
14.3
0.01407
0.01694
0.01342
0.01242
0.01311
0. 01442
0.01425
0. 01338
0.01285
0.01450
0.01708
0.01356
0.01247
0.01340
0. 01472
0.01431
0.01372
0.01300
a &
MR I
Test Date Water Temp. RPM Angle Hgt. V avg. Rn Cd Cd(S)
(OF) (deg) (cm) (cm/sec) ( x 10-5)
12-24-77
"3
"3
"'
12- 25-77
I
"3
"
"5
12-25-77
"
"3
"3
"u
12-26-77
80.0
'3
"3
",
"'
80.0
to
"3
"3
'3
80 0
",
'3
"
"3
80 .0
50
100
150
200
300
50
100
150
200,
300
50
100
150
200
300
50
0.0
"3
",
",
",
2. 0
"3
3'
",
-33
6.0
8.0
3'
",
3,
",
8.0
80-0
11
"0
8.0
99.1-
203.4
304.1
404.2
605.1
101.9
204.4
302.1
400.9
604.5
99.4
203.1
302.7
403.2
603.5
98.7
2o-33
4.79
7.15
9.51
14.2
2.40
4. 81
7.11
9.43.
14.2
2.34
4.78
7.12
9.49
14.2
2.32
0.01652
0.01488
0.01376
0.01313
0.01233
0.01457
0.01416
0.01329
0.01393
0.01297
0.01564
0. 01422
0.01363
-0.01327
0.01331
0.01540
.1 ___________ I ___________ L I.
0.01653
0.01498
0.01392
0.01325
0.01248
0. 01500
0.01436
0. 01375
0.01405
0.01321
0.01632
0.01434
0.01391
0.01343
0.01341
0.01581
a
Table 5.95 Keel No . 1 With two rows of turbulence stimulators
Table 5.5 (continued)
Test Date Water Temp. RPM Angle Hgt. V avg- Rn Cd Cd(S)
(OF) 
- (deg) (cm) (cm/sec) ( x 10-5)
12-26-77
"3
"U
"'
12-26-77
80.0
'
"3
"'
80.0-
100
150
200
300
50
6.0
"'
"o
"'
0.0
8.0
"
",
"3
4.0
201.5
301.0
400.8
602.1
100.0
4.74
7.08
9.43
14.2
2.35
0.01509
0.01422
0.01565
0.01331
0.02001
I * L L A N A
0.01520
0.01453
0.01596
0.01341
0.02037
I
47
6. CONCLUSIONS AND RECOMMENDATIONS
6.1 Experimental Conclusions
As can be seen in the results section, all the numerical values claculated
from the data appeared quite reasonable. In fact, for the first testing done in
the facility using the LDA device in this type of study, the agreement with published
data in Abbot & Von Doenhoff is remarkable. In view of the crudeness of some of
the data reduction, and the possible errors in collecting data it must be assumed
that this device is quite forgiving as regards to its operation. This would tend to
indicate that the LDA can be expected to give extremely accurate results if a
large scale, well planned study is to be done. Conversely, one can quite
possibly develop a technique to obtain drag coefficients to a level suitable for
most non-critical engineering which would be very simple and easy to use.
Other possibilities for use of the instrument not touched upon in this study -
were the use of a frequency spectrum analyzer, and a wake survey with the intent
of calculating lift. As the existance of the turbulent layer was quite visible on the
oscilloscope, and to the LDA's capabilities, this layer constitutes an extensive
region, investigations of the internal motions of wakes and boundary layers do
not appear unreasonable.
Another feature of the device is its comparative freedom from ranges of
greater or lesser accuracy. With the addition of a carriage capable of streamwise
motion, one should be able to measure drag from one to thirty feet per second in
the MIT water tunnel with a constant level of accuracy. This is not possible to
do with any of the presently ava ilable dynamometers.
48
6.2 Recommendations
Assuming that there will be a continued interest in the study of wakes,
boundary layers and flow patterns using the LDA, in view of the excellent
results obtained by all the users of the instrument so far, it is felt that the
following are possible methods for improving the facility.
The present carriage for the laser and the receiving optics was designed
to move only vertically and transversely, as the propeller tunnel has the capa-
bility of moving the propellers axially. In order to check on the assumption that
the pressure variations across the wake are not effecting the survey, the mounting
should have a freedom of motion parallel to the flow in the tunnel. Aside from this
it would also greatly simplify the initial setup of the unit. With this capability
one would only need to align the unit so that it is perpendicular to the flow, with-
out the added difficulty of trying to aim it at a specific location at the same time.
The present system has a handwheel and screw method for trnsverse
motion. The transverse scale used was graduated in millimeters, and readings
and adjustments were done by hand. As the water tunnel will only accept rather
small models, the size of the wakes in this study was typically two or one centi-
meters across. A system that could be adjusted to a tenth of a millimeter would
at times be useful. T he LDA system can easily detect the changes in velocity
that occur over a distance of one millimeter, and for high speed flow.measured
closely behind the keel these changes can be sizable.
It is thought that the system of a pointer and a graduated scale for transverse
position location is not in keeping with the accuracy of the LDA. More important
-19
perhaps is the time required to adjust the position by hand. The time required
for each survey (one transverse at a constant flow) is not that large, approxi-
mately one-half hour. When the time of the data preparation is added on to this
it grows considerably. Any wake survey which was to look at a keel as a whole,
without the simplifying assumption of a two-dimensional section, would require
a very large number of data points. As a result of these two effects, inaccuracy
of positioning the unit, and time for data reduction, it is recommended that for
any extensive study serious consideration be given to the installation of an auto-
mated carriage and data acquisition system. A unit which could traverse the
tunnel, and automatically record the impeller RPM, the velocity of the flow, and
the location (in three dimensions) on some medium capable of direct computer
input would make extensive, and extremely accurate, drag calculations possible.
Without such a system only two-dimensional studies are feasible, if only for the
reason that test time in the water tunnel is not unlimited. As this study found,
the two-dimensional drag is only a fraction of that measured on the dynamometer.
The system should, ideally, have the transmitting and receiving optics
mounted on the same baseplate. This would help eliminate noise in the system
caused by vibration, and misalignment between the two. Mis-ligned optics were
probably the greatest cause of delay in the testing.
The system should also include some system for establishing a reference
point with the tunnel or the model. In this study the aft end of the model was
adequate as a reference point, but on other shapes that point might not be.
The plexiglass walls on the test section leave a great deal to be desired as
50
far as use with this instrument. Any imperfections on the tansmitting side intro-
duces as error in the data that could not be reasonably detected or corrected.
The present method of mounting the models on the dynamometer and
splitter plate is fairly cumbersome. It has the advantage that one can compare
the drag measured with the dynamometer and the wake survey directly. With
the dynamometer and splitter plate setting up the system takes a good deal of
time, as does changing from one model to the next. The present mounting also
restricts the LDA to the bottom two-thirds of the keel. For these reasons it is
recommended that any new work with this equipment be preceeded by the design and
installation of a n ew splitter-plate or mounting body that would permit more
rapid changes of models, and give an unrestricted view of them to the LDA.
51
REFERENCES
Abbott, Ira H. and Von Doenhoff, Albert E. : "Theory of Wing Sections", Dover
Publications, Inc. , New York, 1949.
Goett, Harry J.: Experimental Investigation of the Momentum Method for
Determining Profile Drag. NACA Rept. No. 660, 1938.
Kerwin, Justin E. and Lewis, Dean: Water Tunnel Tests of a Series of Yacht
Keels with Varying Reynolds Number. Unpublished Report for SNA ME,
panel H-13.
Milne-Thomson, L. M., C. B. E.: "Theoretical Aerodynamics", Dover Publica-
tions, In., New York, 1958.
Schlichting, Dr. Hermann: "Boundary- Layer Theory", McGraw-Hill Book
Company, 1968.
APPENDIX I
Data Reduction
The data reduction program that was used for this report is based on the
fit of a poynomial to the experimental wake velocity survey. There were two
variations of this program, the one listed herein was used to calculate the
results. This version was probably slightly less rigorous than the other, but it
was much easier to prepare the input data for it. This program also relies less
on the assumption that you can get a good fit with a polynomial to the wake survey
data.
The program requires input of the date of the test run, the water tunnel
impeller RPM, the angle of attack, and the height of the laser above the tip
of the keel. These values served merely to identify the computer run and the
data at a later time. The approximate center of the wake is required. This value
is input so that the curve fitting routine will have the data roughly centered, and
so give a better fit with the polynomial. This value was usually obtained by
eye from the graph of the wake survey data.
The actual data of wake velocity, transverse posistion,was split into two
groups. These were the data points inside the visible wake region, and the free-
stream flow outside the wake. Generally these were also obtained from the -
graph. Obviously, this is a point where different operators will make different
choices as to the extent of the wake region. The program fits a separate polynomial
to each of these data groups. The degree of the polynomial used is also left up to
the operator, with an upper limit of a ninth order term. For the fairly smooth
flows that can be assumed to exist outside the wake a low order polynomial is
all that should be required. As the function for the drag is based on the difference
5) 3
I. (continued)
between these two polynomials, it is felt that the lower the order of the polynomial
that can be used and still represent the flow the better. Higher order polynomials
tend to introduce a number of "humps and hollows" which would probably not exist
in a real flow. Once the polynomials are established, the value of the integrand
in the drag function is calculated for each of the points in the wake. The drag
function for a keel with span "b" is;
DpS U-U,(UU 6) IA
Ua is the extrapolated free stream flow, Ub is the viscous flow, and Us
is the free stream flow at the edge of the wake region. A polynomial is fitted to
this function and integrated between the limits of the wake region to calculate the
drag. The program calculates the two-dimensional section drag coeficient based
on the chord of the keel section. The value of the chord was a data point inside
the program, as is the programs conversion from volts to cm/sec. As all of
the keels tested had the same chord, and used the same optics and settings on
the tracker this was a convenient method. The values are on lines twenty and
twenty-one of the program.
The second variant of the program operated in exactly the same manner as
the first, with the exception of the extent of the region of the wake polynomial.
Here one polynomial is fitted to the free-stream points, and the other is fitted
to all of the data . If a good fit can be obtained to the data by the polynomials
then once outside the wake they should be substantially the same. Thus the end
points chosen by the operator should make little difference to the integral, and
so do not require the care in their choosing that the other variant should have.
5r
I. (continued)
The difficulty encountered with this program was that if required more keypunching
of input data, or the development of a routine that could recognize a "wake". As
the function being integrated has comparatively very small values at the points
where the wake and the undisturbed flow intersect, it is felt that the operator
estimates involved in the first variation were adequate for the purposes of this
work. In view of the very small forces and low drag coeficients being measured,
small errors that might be induced by the operator are probably overshadowed by
other physical effects. These effects might be scale effects, tip effects,
imperfections on the optical surfaces, and calibration defects in the electronics.
The following is a table of the variation of the calculated drag coeficient as the
operator parameters were varied over a wide range. The table is based on the
test data of 11-13-77, Keel No. One, at two degrees angle of attack , the survey
being done eight centimeters above the tip of the model.
Degree of Wake Degree of Free- Uavg. Cd Cd (S)
polynomial Stream polynomial
8 2 205.8 0.007140 0.007169
3 2 205.8 0.007398 0.007427
8 6 205.6 0.006972 0.007084
4 2 205.8 0.007177 0.007206
6, 5 205.7 0.007146 0.007197
Uavg. is in centimeters per second, and is the calculated velocity at
the edge of the wake based on the free-stream polynomial. This value was used
to determine the Reynolds Number. Thus, over a wide range of parameters the
drag coeficient is essentially constant at a value of 0. 0071, and as most engineering
practice would use only two digits, this method should be adequate.
55
APPENDIX II
Effect of a body on wake velocity in an ideal fluid using a Rankine Ovoid to
simulate the keel.
We are assuming the existance of an ideal two-dimensional fluid flow. The
wake survey is done through a line some distance behind a body. Modeling the
body, in this case a keel, as a Rankine Ovoid we will investigate the effect the
body has on the Ovoid so that it has the same thickness as the keel models, and
the same overall length, and lies on the "X" axis. As this is not nearly as
streamlined a body as our keels it should exaggerate the difference in velocity
in the wake.
A
We will consider a two-dimensional body of length 2L and of breadth 2B,
with a source located at -A on the X axis, and a sink at +A on the X axis in a
free stream flow of U. The formula for the velocity around the ovoid thus
formed is: + 4(X+0A) -/I (X -A)
T21((x+A) 2 + y<) 2 (rX-A) 2 , y 2)
At X=L, and Y=O. 0 there should be a stagnation point and so at this point u would
equal zero. Substituting these values and rearranging terms yields:
21 (-A) 2f(1A
We know the flow inside the body must equal the strength of the source M. This
h
yields: H -_ , M(xA)_ - M x-A) Cy
22" (xrA)zt ye) 2'((x-A)+y')
56
Which after integrating and rearranging terms becomes:
/ ~-TAM' 8  t /-r j-/
28 21n A d2fbA
U
Equating these two terms for M, and substituting our values of L and B of 10. 16
U
and 1.524 cm respectively we can solve numerically and obtain a value for M
of 0.2953203.
For our test case of 100 RPM at 2 degrees angle of attack of 11/13/77,
where U = 205. 7 cm/sec., M equals 696. 5319. As the keel is swept back at
40 degrees, and we are 8 cm above the tip the distance behind the keel is 6.7128 cm.
Substituting these values into the formula for velocity u, the difference in velocity
across a typical half width of a wake of one crr is about 0. 3 cm/sec. or less than
two tenths of one percent of the flow speed. As this was for a "worst-case"
body it was believed that at this distance behind the keel the pressure terms
could be neglected.
57
APPENDIX III
Program Listing
C PROGRAM TO REDUCE T-lE LASER DPPLER ANEMOMETER WAKE SURVEY
C TO A DRAG COEFICIENT
C
C PROGRAM WRITTEN BY JOHN KNOBEL ROCM 5-330B COURSE 13
C
DIMENSION XW(40),VW(40),UW(40),CW(40)
DIMENSION XFS (20) ,VFS (20) ,UFS(20) ,CFS(20)
DIMENSION D(40),CD(10),DS(40),CDS(10)
INTEGER MOsAYYRRPMALPHAHGT
C
C X IS THE TRANSVERSE DIMENSION IN CM
C V IS THE ADJUSTED VOLTAGE PEALING
C U IS THE VELOCITY
C C IS THE CCEFICIENT OF THi PCLYNCMIALS
C IN ALL CASES THE SUFFIX 'w' REFERS TC WAKE ELGICN
C AND 'FS' REFERS TO THE FREE STREAM
C
C NC IS THE NUMBER CF COEFICIENTS IN THE POLUNOMIAL
C CU IS THE VELOCITY CALCUIATID FROM TUE POLYNOMIAL
C VTOC IS THE VCLTS TO CY/SEC CONVERSION
VTOC=408.7
CHORD=20.32
C
C
C 333 IS THE PESTAPT FCINT FC? AEDITIONAL DATA
333 CONTINUE
C
C SFT ALL ARRAYS TO INITIAL VALUE ZERC
CALL ERASE (XW,20,VW,2C,UW,20,CW,20)
CALL ERASE (XFS,20,VFS,20,UW,20,CW,20,D,20,CD,1C)
C
DATA INPUT SLCT.ON
o SET U? AN ID SYSTEM FOR DATA
C FIEST 312 FCP VONTH DAY YEAR, THEN 2X THENI3 FOR RPM 2X
C THEN 12 FOER ANGLE OF ATTACh AND 2X AND 12 FCR HT AECVE TIP
READ(5,310) MO,DAY,YF,,PM4ALPHA,iGT
LDA 10001
LDA10002
LDA10003
LDA10004
LDA10005
LDA10006
LDA 10007
LDA1000d
LDA 10009
LDA10010
LDA 10011
LDA 10012
LEA10013
LDA 1001 4
LLA10015
LDA 10016
LDA10017
LDA 10018
LDA10019
LDA10020
LDA10021
LUA10022
LDA 10023
LDA10024
LDA10025
LDA10026
LDA10027
LDA10028
LDA10029
LDA1003C
LDA 1003 1
LDA10032
LDAl0033
LDA10034
LEA10035
LDA10036
CC
C
C
C
"C
C
C
C
C
C
C
C
C
11 CCNTiNUIE
102 FCF71AT(2F12.6)
ECHO OF INPUT EATA
310 FOR-tAT(3I2,2XI3,X,12,2XI2)
PEAD(5,103) CL
103 FORMAT (F12.6)
CL IS THE APPROXIMATE CENTER OF THZ WAKE
IT IS INPUT IN AN ATTEMPT TC GET A BETTER CURVE FIT 01/25/78
DTATA FOR THE WAKE AND FREE STREAM IS INPUT SEPARTATELY
THE WAKE VALUES ARE ALWAYS REAE FIRST
VALUES OF XW MUST FE ENETERED SEQUENTIALLY
READ (5,101) NWNCW
101 FCRMAT(12,2X,12)
DO 10 i=1,NW
FEAD (5, 102) X W (I) ,VW(I)
XW (I)=XW (1)-CL
IW (I) =VW (I) *VTCC
IC CONTINUE
PEAD (5,101)NFS,NCFS
DC 11 1=1,NFS
EEAI(5,102) XFS(1),VFS(I)
XFS (I)=XFS (1) -CL
UIFS(I)=VFS(I) *VTOC
I
LDA10037
LDA10038
LDA10039
LDA10040
LDA 100141
LDA10042
LDA10043
LDA10044
LDA10045
LDA1C046
LDA10047
LDA10048
LDA 10049
LDA10050
LDA 10051
LDA10052
LDA10053
LDA10054
LDA10055
LDA10056
LDA10057
LDA 10058
LDA10059
LDA10060
LDA 10061
LDA10062
LDA10063
LEAl0064
LDA10065
LDA10066
LDA10067
LDA10068
LDAl0069
LDA10070
LDA10071
LDA 10072
C
C
C
WPITE (6,208) MO,DAY,YRRPM,ALPHA,HGT
208 FORMAT(2X,' DAlE AND RUN ',616)
WRITE(6,200)
200 FORMAT (10X,'THIS IS THE LDA WAKE SURVEY DATA REDUCTION')
WRITE(6,313) NWNCWNFSNCFS
313 FORMAT('NWNCW= ',214,' NFS,NCFS= ',214/)
WRITE(6,201)
201 FCRMAT(10X,'XFS',7X,'VFS',7X,'UFS')
DO 41 I=1,NFS
WRIT E(6,202) XFS (I),VFS (I),U F S(I)
202 FORMAT(3F12.6)
41 CONTINUE
WRITE(6,203)
203 FORMAT (1OX,'XW',7X,'VW',7X,'UW')
DO 42 I=1,NW
W RIT E(6,v20 2) XW (1),VW (I),vUW (I)
42 CONTINUE
WRITE(6,207) Cl
207 FORMAT('XW=XDAIA-CL, CL= ',F12.6//)
CC
C FIT A POLYNOMIAL THFU THE WAKE AND FREE STREAN REGIONS
C A SEPARATE POLYNCMIAl FCR EACH
C
CALL LSFIT(NFSNCFSXFSUFS,CFS)
C
CALL LSFIT(NW,NCWXWiUW,CW)
C CALCULATE THE US FCR USE IN THE MGMENTf1M TRANSFER
AVGFS1=CFS (1)
AVGFS2=CFS(1)
C AVFRAGE FREE STREAZ VELOCITY = AVGFS
DC 40 I=2,NCES
C
12=1- 1
AVGFS1=CFS(I)= (XW (1).I2)+ AVGFS1
LDA10073
LDA10074
LDA10075
LEA10076
LDA10077
LDA10078
LDA10079
LDA1C080
LDA 10081
LDA10082
LDA 10083
LDA10084
LDA10085
LDA10086
LDA10087
LEAl0088
LDA10089
LDA1 0090
LDA10091
LDA10092
LDA10093
LDA10094
LDA10095
LLA1009b
LDA10097
LDA1G098
LDA10099
LDA10100
LDA10101
LDA 10102
L D A10 10 3
LDA1 C104
LDAlo105
LDAl lb
LDA 10107
LDAl10136
C
AVGFS2=CFS (1) * (XW (NW) **I2) +AVGFS2
C
40 CONTINUE
AVGFS=(AVGFS1+ AVGFS2)/2.0
C
C
C CALCULATE, AT EACH WAKE POiNT, THE VELOCITY PBEDICTED BYT
C THE FREE STRIAP CURVE FIT
C
WRITE (6,210)
210 FORMAT(5X,'AT XW, THESE ARE THE VALUES OF CUW,CUFS')
C
DC 20 I=1,NW
C
CUFS=CFS (1)
CUW=CW(1)
C
DC 21 K=2,NCFS
C
K2=K-1
CUFS=CFS (K) * (XW (I) **K2) +CUFS
C
21 CONTINUE
C
C CALCULATE WAKE VELOCITY FROM THE POLYNCIAL
C
DC 22 K=2,NCW
K2=K-1
CUW=CW (K)I (XW (I) **K2) +C[JW
C
22 CONTINUE
C
WPITE(6,211) XW (I),CUJW,CUFS
211 FCPrAT(4X,F12.6,4X,F12.6,4X,F12.6)
C
D (I) = (CUFS* (CU6S-AVGFS))-(CUW*(CUW-AVGFS))
LDA10109
LDA10110
LDA10111
LDA10112
LDA10 113
LDA10114
LEA10 115
LDA10116
LDA10117
LDA10118
LDA10119
LEA 10 12C
LDA 10121
LDA10122
LDA 10123
LDA10124
LDA 10125
LDA 10126
LEA10127
LDA10128
LDA10129
LDA10130
LDA10131
LDA10132
LEA10133
LDA10134
LDA10135
LDA10136
LDA1O137
tEA1 C138
LDA 10139
LDA1r140
LDA10 141
LDA10142
LEA10143
LDA 10144
DS (I) =CUW* (AVGFS-CUW)
20 CCNTINUE
C
C
C
C
C
C
C
C
C
C
WRITE(6,212)
212 FCRMATi(//)
D(I) IS NOW THE EATA SET FOR THE CURVE OF X VERSUS U(UFS-U)
JAN 22 DRAG TERM IS 01**2-U2**2-US(U1-U2)
WE CAN INTEGRATE THIS TO GET THE DRAG
CALL LSFIT(NWNCWXWD,CE)
THE POLYNOMIAL IS INTEGRATED TO GET THE DRAG
THE WAKE END PCINTS ARE THE LIMITS FOR THE INTEGRAL
DRAG COEFICIENT = DC
DC=0.0
DC1=0.0
DC2=0.0
DO 30 I=1,NCW
DC=CD(I)* ((W (1) **I)/I) +DC1
DC2=CD(I)*((XW(NW)**I)/I)+EC2
30 CONTINUE
DC=2. 0* (DC2-DC1) /(AVGFS*AVGFSxCHCRD)
C
C CALCUTATION OF SIMPiIFIED EAG CCEFICIFNT
CALL LSFlT(NW,NCW,XW,D.,CDS)
Ds2C=3.0
DSC1=U.
DSC2=.0
C
I
LDA10145
LDA 10146
LDA1O147
LDA10148
LDA 10 149
LDA10150
LDA 10151
LDA10152
LDA10153
LDA10154
LDA 10155
LDA 10156
LDA10157
LEA 10158
LDA10159
LEA1C160
LDA 10161
LDA10162
LDA10163
LJEA10164
LDA 10 165
LDA10166
LDA10167
LDA10168
LDA10 169
LDA 10170
LDA10171
LEA 10172
LDA 10173
tEAl0174
LDA 10175
LDA10176
LDA 10 177
LDA10178
LDA 10179
LDA10183
C
C
C
C
C
I
DO 32 I=1,NCW
DSC1=CDS (I)*((XW (1) **I) /1) +DSC1
DSC2=CDS (I)* ( (XW (NW)**I)/I)+DSC2
32 CONTINUE
DSC=2. OlvDSC1-DSC2) / (AVGFS*AVGFS*CHOP D)
OUTPUT SECTION
WHITE(6,209) AVGFS
209 FORMAT (5X,'AVGFS AT WAKE BCUNJ2ARY= ',F12.6/)
C
C
C
C
C
C
C
C
C
c
PESTART SECTICN
READ(5,300) INDEX
300 FORM AT (12) l
IF (INDEX.EQ. 1.0) GO TO 333
END
LDA10181
LDA 10182
LDA10183
L EA 10184
LDA10185
LDA10186
LDA 10187
LDA10188
LEA 10189
LDA10190
LDA10191
LDA10192
LDA10193
LDA 10 19 4
LDA10195
LDA10196
LDA 10197
LDA10 198
LDA10199
LDA10200
LDA 1020 1
LDA1 0202
LDA10203
LDA1020 4
LDAlC205
LDA10206
LDA10207
LDA10208
LEA10209
LDA 10210
LDA1021 1
LDA10212
LDA10213
LEA 10214
WRITE (6,204)
204 FORMAT (5X,'COEFICIFNTS CF FSWAKE')
DO 43 I=1,NCFS
WRITE(6,205) CFS()
43 CONTINUE
DO 44 I=1,NCW
WRITE(6,205) CW(I)
44 CONTINUE
205 FCRMAT(5X,E12.5)
WRITE(6,206) ECLSC
206 FORMAT(5X,'DRAG CCEF =',F12.6,' SIMPLIFIED DC =',F12.6)
WRITF(6,312)
312 FCRMAT (iH1)
APPENDIX IV
EQUIPMENT IST
1. "Laser Doppler Anemometer Signal Processor", (Tracker), Thermo-Systems
Inc. Model 1090
2. "LDA Frequency Shifter", Thermo-Systems Inc. , Model 985
3. Laser, Helium-Neon, Spectrophysics Model 124A
4. "Spectra-Physics Laser Exciter", (Laser Power Supply)
5. "Photo-multiplier", (Photo-detector), Thermo-Systems Inc. Model 962, Optics
310 mm focal length.
6. "Photo-multiplier Power Supply", Thermo-Systems Inc. , Model 965
7. "Signal Conditioner", Thermo-Systems Inc., Model 1057
8. Oscilloscope, Tektronax,Model 561B
9. Frequency Counters, Hewlett Packard,Models 5381A & 5321B
10. Voltage-to-Frequency Converter, Dymec Model,2210
11. Bragg Cell and Optics, Thermo-Systems Inc., Model 976, Focal Length
309 mm.