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EDITORIAL COpy
UNIVERSlTYOF MINNESOTA
ST. ANTHONY FALLS HYDRAULIC LABORATORY
Technical Paper No. 46, Series B
Experim.ents on a Jet Flap in Supercavitating Flow
by
EDWARD SILBERMAN
Research Conducted for DAVID TAYLOR MODEL BASIN
Department of the Navy Washington, D.C.
under Bureau of Ships Fundamental Hydromechanics Research Program
SR-009-01-01 Office of Naval Research Contract Nonr 7lO(50)
January 1964
Minneapolis, Minnesota
UNIVERSITY OF MINNESOTA
ST. ANTHONY FALLS HYDRAULIC LABORATORY
Technical Paper No. 46, Series B
Experim.ents on a Jet Flap in Supercavitating Flow
by
EDWARD SILBERMAN
Research Conducted for DAVID TAYLOR MODEL BASIN
Department of the Navy Washington, D.C.
under Bureau of Ships Fundamental Hydromechanics Research Program
SR-009-01-01 Office of Naval Research Contract Nonr 710(50)
Janu,ary 1964
Minneapolis. Minnesota
Reproduction in whole or in part is permitted
for any purpose of the United States Government
Experim.ents on a fully-cavitated, flat-plate hydrofoil equipped w;ith
a pure jet flap were conducted in the free-jet water tunnel at the st. Anthony
Falls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and
were tested in 6-in., 10-in., and l4-in. wide free jets. Data were obtained
on the increments in lift, drag, and moment coefficients and on the shift in
center of pressure as a function of jet momentum coefficient C., cavitation J
number, and other variables.
The experim.ental results were qualitatively similar to those that
are obtained in fully wetted flow. The increment in lift, for example, is
given by
where k = 1.27 in two-dim.ensional flow near zero cavitation number for jet
flap angles T = 36, 38, 50, and 60 degrees and angle of attack of 8 or 11
degrees. The coefficient k was 1.58 for T = 20 degrees and decreased very
slightly with increase in cavitation number at all values of T. These values
of k are considerably less than the value given by linear theory which is
about 2.10. For an aspect ratio 2.5 foil, k was about 1.10 experim.entally.
Jet thrust recovery is less than unity, the value to be expected
.from theory and from wetted flow experim.ents, and decreases as both flap
angle and angle of attack increasec Jet flaps may be particularly useful in
controlling unwanted ventilation of a hydrofoil operating near a free surface.
iii
CON TEN T S -- ... --~--Abstract • • • • • • • • • • • • • • • • • • • .. . . . . . . . . .. List of Illustrations •••• • • • • • • • • · .. .. . . . . . . .
I. INTRODUCTION . . .. . • • • · .. ., . .. . . . • • • • • • • • •
II. EXPER:rnENTAL EQUIPMENT •• · . . .. . . . .. . . . . .. . . . . III. PROCEDURE AND DA.TA • • • • • • .. .. ., .
• lit • .. • • • • • • •
IV. SOME OBSERVATIONS OF THE FLOW /II • • • .. • , • .. • • .. • • lit
V. DISCUSSIONS OF NUMERICAL DATA • • • • • • • • • · • • • • • A. Two-dimensional Experiments, Cj = 0 • · • • • • • • B. Two-dimensional Experiments, C. .> 0 • • • • • • • • J C. Thrust Recovery • • • • • • • • • • • • • • • • • • D. Ventilated Cavities with Jet Flap • • • • • • • • • E. Effect of Submergence • • • • · • • • • • • • • • • F. Finite Span Experiments • • • • • • • • • • • • • •
VI. SUMMARY • . • • • • • • • • . • • • • • • • · • • · • • • •
List of, References • • • • • • • • • • • • • • • • • • • • • • • • Figures 1 through 25 •• •••••••••••••••••••• Appendix A Notation • • • • • • • • • • • • • • • • • • • • • • Figure Al ..... . • .. . .. • • . . . .. . lit .. • • • • .. • .. .. .. .. •
Appendix B The Jet.f1ap Dynamometer • • • • • • • • • • • • • • Figures Bl through:82 • • • • • • • • • • • • • • • • • • • • • • Appendix C - Effect of Free Jet Deflection • • • • • • • • • • • •
iv
Page iii
v
1
2
10
12 12
14 17 18 22 2;3
24
29 ;32 60 61 65 70 75
Figure
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
The Two-dimensional Free-jet Water Tunnel • •
Front Face of Tunnel . . . . The Dynamometer and Two Hydrofoils . . Construction Details, Jet-flapped Hydrofoils . . . Typical Super-cavity without and with a Jet Flap . . . Cavities Produced by a Two-dimensional, Jet-flapped Hydrofoil
Cavities Produced by a Finite-span" Jet-flapped Hydrofoil
Lift Coefficient as a Function of Cavitation Number, C .. = 0, fft= 00 III ••••••••••••••••••
J
Drag Coefficient as a Function of Cavitation Number, C. = 0, JSft= 00 .................... .
J
Moment Coefficient as a Function of Cavitation Number, C. = 0, JSIl= CD ••••••••••••••••••••
J
Center of Pressure as a Function of Cavitation Number, C. = 0, .fffl= CJ) • • • • • • • • •• • •••
J
Typical Force and Moment Coefficients, C. > ° J
Increment in Lift Coefficient near 0" = ° Increment in Lift Coefficient •
Increment.in Drag Coefficient ••
Increment in M0ment Coefficient •
Increment in Center of Pressure
Increments in Coefficients for Small Submergence
Lift Coefficient as a Function of Cavitation Number, C. = 0, m= 2.5 . . . . . . . . . . . . . . .
J
Drag Coefficient as a Function of Cavitation Number, C. = 0, m= 2.5 . . . . . . . . . . . . . .
J
.
. Moment Coeffic-ient as a Function of Cavitation Number,
.
.
c. = 0, .lB.= 2.5 . . . . . . .... J
Increment in Lift Coefficient, m= 2.5 •
v
.
.
. .
. .
Page
32
33
34
36
37
38
39
40
41
42
43
44
45
46
48
50
52
54
55
55
55
56
Figure
23
24
25
A-l
B-1
B-2
Increment in Drag Coefficient, JR = 2.5
Increment in Moment Coefficient, JR = 2.5
Thrust Recovery
Definition Sketch
. . .. . . . . .
Typical Calibration Curve for Dynamometer
Sample Record from Rear Dynamometer Unit • •
vi
. ". . "
" . . " . ..
Page
56
56
57
61
70
71
I!~I!!~I!I~ Q! A ill l~!~ !li ~Qfl!Q!!!I!I!liG E~Qli
I. INTRODUCTION
The jet flap, like the solid flap, is a device for oontrolling the
lift of a.n airfoil or hydrofoil independently of incidence. The jet flap
principle is especially attractive for application to a vehicle using a jet
propulsion system, although the jet may be independently produced for. the
purpose of controlling lift, as it is in model studies. The jet fluid is
made to is.sue at high velocity from a slit at the trailing edge of the foil,
the angle of emergence with respect to the foil being controllable. Lift is
attributable to the sum of the direct reaction of the jet and the net pres
sure force on the exterior of the foil createp. by ~he flow pattern. For a
foil at fiXed incidence the jet flap increases the lift over the case without
flap through both of these components, and the role played by the second is
usually larger than that played by the first. The change in flow pattern al
so influences the drag, while the ·emitted jet· produces thrust.
For fully wetted flow a large literature on the jet flap has devel
oped. This is reviewed and the most important references are given in several
* articles in the book edited by Lachmann [1]. A detailed technical review
maybe found in the paper by Williams, Butler and Wood [2J.
The most useful theoretical results have been obtained through use
of linearized thin-airfoil theory. They are well supported by experimental
data,even beyond what might be considered the range of validity of the lin-
** ear theory. In general, it is found that for small jet momentum coefficient
Cj and small angle of attack, the increase in lift coefficient attributable
to the jet nap 6CL is
6CL = kvc.- sin T . J (1)
where T is the angle of jet emission measured with respect ,to the foil chord
and k is a proportionality constant which decreases with aspect ratio. The
* Numbers in brackets ref~r to the List of References on page 29.
** Notation is defined in Appendix A.
2
full reaction of the jet (not just the component in the flight direction)
appears as thr~st in the ideal case, and this is nearly realized in practice
for 'T' less than 60 degrees. The center of lift on a foil moves well to the
rear with the flap operating, and the nose down moment increases with C .• J
The jet flap principle is also applicable to hydrofoils operating
under super cavitating conditions. Inthis application the alteration of flow
pattern produced by the jet flap is confined essentially to the wetted sur
face of the foil, the other surface being open to the cavity. For this reason
it may be expected that ACL for a given jet flap will be less than in the
fully wetted condition. Also, a much reduced influence on drag should be
expected, while the full jet momentum should still be recoverable, ideally.
A super cavitating hydrofoil operating with jet flap may be seen in Fig. 5b.
For the super cavitating case linear thin-airfoil theory has been
applied by Ho C3J to the two-dimensional, flat-plate hydrofoil at zero cavita
tion number. The results cannot be expressed in such simple form as for the
fully wetted case. However, within the range of small C. of interest in J
this report, analysis of Ho's numerical results indicates that the increment
in lift is given roughly be Eq. (1) with k~2.10 for small T and angle of
attack (see Fig. 13). This compares with a k-value of the order of 2 vTf ~ 3.5 for the corresponding fully wetted case [lJ. The calculated increment
in drag coefficient also increases with T,
the center of pressure moves to the rear as
but not so rapidly as lift, and
C. increases. J
Nothing has appeared in the literature regarding experimental work
on the jet flap in super cavitating flow. It is the objective of this paper
to report on such experimental work on a flat-plate hydro'foil.
II. EXPERIMENTAL EQUIPMENT
These experiments have been conducted. in the two-dimensional test
section of the free~Jet water tunnel at the st. Anthony Falls Hydraulic Lab
oratory [4J. The tunnel is designed to operate over a range of cavitation
numbers down to zero~ Water is dra-wn from and wasted to the MissiSSippi River
so that there is no problem in disposing of the jet flap fluid or of air bub
bles generated by cavitation. Thewidth of the free jet is variable, and the
jet posi;tion can be controlled to place a test body on or off the centerline
3
of the tunnelo The test section is transparent to promote visualization and
photography. Figure 1 is an over-all photograph of the tunnel te.st section
area, ~hile Fig. 2 shows the front face of the tunnel operating with a super
cavitating hydrofoil in the test sectiob~ The tunnel is provided with cali_
brated, manometers for reading test stream velocity, ambient pressure surround
ing the free jet, and cavity pre'ssure within a cavity.
Flat-plate hydrofoils of 2-in. and 2.75-in. chord have been used in
the experiments. The majority of the tests were conducted with two-dimen
sional foils of 2-in. chord spanning the 5-in. thickness of the free jet.
Some experiments were also conducted with foils of 2-1/2-in. span (and 2-in.
chord) extending inward from one face of the tunnel and spanning only half
the jet (aspect ratio == 2.5). The foils are actually wedges of about 10.5
degrees central angle, but when a wedge is installed in the tunnel with its
axis sufficiently off the 'flow axis, one surface falls inside the cavity cre.
ated by the flow while the other becomes the flat plate under test. The base
of each wedge consisted of a slit tube running the length of the span; the
jet flap was emitted from the slit which was located at the trailing edge of
the foil on the wetted side. Two of the foils m.9.Y be seen in the photographs
in Fig. 3, while Fig. 4 shows some typical construction details. The chord
length c of each foil was measured from the leading edge to the slit. The
flap angle T was not adjustable; consequently otherwise identical foils with
different flap angles had to be provided. Table I lists the foils that were
available and tabulates some of their properties.
TABLE I
PROPERTIES OF JET-FLAPPED HYDROFOILS
Center of Dynamometer Nominal Jet
Foil Aspect Chord, Flap Angle, from Mid-chord Slit Width Thickness No. Ratio c-in. T--deg. m--in. in. o--in.
1 (J) 2.0 20 0.194 0.010 0.01043
2 (J) 2.0 36 0.246 0.012 0.01245
.3 (J) 2.0 60 0.226 0.015 0.01875
4 2.5 2.0 33 0.246 0.013 0.0120
5 (J) 2.75 38 0.619 0.012 0.0121
6 (J) 2.75 50 0.619 O.Oll 0.01155
4
The tunnel was provided with a specially designed dynamometer system
to permit activating the jet flap without interfering with measurements. The
system consisted of two identica~ units, one on the front face and one on the
rear face of the test section. The front face dynamometer is shown in Fig. 2
while Fig. :3 shows some details of the dynamometer and the method of attach ...
ing test bodies.
The dynamometer units are of the displacement type. Small displace
ments are measured by use of strain gages fastened ~o fibers supporting beams
outside the tunnel. The strain gage bridges are balanced, and output voltages
are read as time averages from dials on carrier amplifiers. Readout consisted , "
':;L of one~-0.rag reading D and two partial lift readings L Land L U for
each face; the latter permitted calculation of pitching moment. The primed , symbols indicate forces along the dynamometer axes, the D -axis lying in the
plane of the beams. Angle of attack was set by rotating the dynamometer; it t
was measured by reading the angle between the D -axis and the vertical, and
subtracting half the wedge angle. True drag and lift were obtained by reso
lution from the measured values. (Although readings from one face would have
been sufficient for lift and drag, the four-point method of support shown in
Fig. :3 for two-dimensional bodies distributed the moment unequally between
the two faces, and readings from both faces were necessary to get true mo
ment.) The dynamometer system and the method of using it are described in
greater detail in Appendix B.
Jet-flap fluid was admitted to a foil through a large hose on each.
face like the one visible in Fig. 2. Water from the city mains boosted by a
pump to a maximum of 150 psi was used for this purpose. It was necessary to
filter the city water in order to keep the slits from clogging. Flow ra te
was adjusted and measured independently for each face; measurement was by
means of calibrated orifices whose pressure drops were read on the 50-in.
manometer visible in Fig. 2. When two-dimensional foils were used, the valves
were adjusted so that the flow was equally divided between the two faces.
The input pressure to the jet flap was also read, but this was used only to
establish the approximate flow rate.
Each hydrofoil shown in Table I was calibrated prior to use to
determine flap angle T and jet thickness o. Calibration was accomplished
by running the flap in the tunnel without operatingthetunnel--that is, with
5
a water jet flap in.air. Lift and drag were measured by the dynamometer for
each of several jet-flap flow rates. The resulting force vector gave the di
rection of the jet flap while the force magnitude gave the momentum of the
jet, and this, together with the measured flow rate, gave the jet velocity
and thickness. The flap angle was also measured by protractor by sighting
through the tunnel. Table I lists the average calibrated values of 'T' and
o at the higher jet flow rates. Extreme deviations in 0 varied from about
5 per cent higher to about 10 per cent lower than the cited averages, while
deviations in 'T' were between 1 degree higher and 2 degrees lower. The aver
age values given in the table have been used in all further computations.
When the tunnel was operating, one side of the jet flap was in con
tact with the flowing tunnel water while the other was in contact with the
water vapor in the cavity. During calibration both sides were in contact with
air. It has been arbitrarily assumed that the jet properties were the same
during operation as during calibration, but this point is discussed further
in Section V, Part C, following.
III. PROCEDURE AND DATA
Two types of experimental data were obtained--photographs. of the
cavities and numerical data read from the various dials and gages. The data
taking procedure (after all calibrations were completed) may be outlined in
the following steps:
(a) A hydrofoil was installed in the tunnel at a given angle
of attack. With the tunnel not operating, jet-flap fluid
was allowed to run through the dynamometer units for a
time in order to stabilize the temperature. The jet-flap
flow ·was then reduced to a trickle and the amplifiers were
zeroed.
(b) The tunnel was started and the flow was adjusted to produce
a cavity of given length (with the jet-flap fluid still at
a trickle.). Tunnel speed U, ambient pressure in the
space surrounding the free jet p , the difference be-CD
tween ambient pressure and cavity pressure p - p (presCD c
sures were read as averages on mercury manometers) , cavity , , length t, and the dynamometer readings D, L U' a.nd
6
, L L from each unit were read and recorded. (During cali-
bration it was demo:1strated that the trickle of jet-flap
fluid through the apparatus affected neither cavity prop
erties nor dynamometer readings except to stabilize the
dynamometer readings.)
(c) The jet flap was then turned on, and the readings tabulated
in (b), above, were repeated. In addition, the jet-flap
flow manometer was read for each face, and the total flow
rate Q as obtained from calibration curves was recorded.
(d) The readings in (c) were repeated for a number of jet-flap
':'l~ flow rates until the maximum capacity of the system was
reached. A final set of readings was then obtained with
the jet flap reduced to a trickle again; if these did not
closely reproduce the data from (b), above, the data were
discarded and the experiment re-run.
(e) The amplifier zeros were checked as in step (a). If the
zeros shifted, the data were also thrown out and the exp
eriment was re-run. The satisfactory completion of this
step marked the completion of one run.
(f) steps (a) through (e) were then repeated for several dif
ferent cavity lengths and tunnel speeds, each set of con
ditions yielding data for an additional run. The majority + of the tests were run at tunnel speeds of 40 - 4 fps, al-
though some were run at speeds as low as 28 fps. Cavity
lengths varied from about 4 to 20 inches and then to in
finity as the cavity "split" open to the atmosphere sur
rounding the jet.
(g) Next, a new angle of attack was set and steps (a) through
(f) were 'repeated; the same procedure was followed for all
the foils listed in Table I. Basically, three nominal
angles of attack were used--8 degrees (the smallest angle
at which the 2-in. chord wedges could be operated continu
ously with a super cavity without the aid of a jet flap),
11 degrees, and 20 degrees. These were supplemented by
~n1ng a few tests on each foil to determine the minimum
angle of attack attainable with the aid of a jet flap at
maximum C. available; for the2-in. chord foils this was J
5 degrees for the foil with T = 60 degrees and 6 degrees
for the others.
(h) After th~ previous program was completed, a few similar
runs were conducted using ventilated cavities with foils
No. 2 and 3. step (b) was first completed without ven
tilation. Then (b) was repeated adding air to the initi-
al cavity; the air flow rate
calibrated float-type meter).
QA was measured (using a
Then step ( c) was accom-
plished while the cavity was still being ventilated at the
same rateo Finally, the ventilation was discontinued, and
(c) was repeated at the same jet-flap flow rate.
(i) Most of the experiments were conducted in a 10-in.-wide
free jet. Some experiments on foil No. 3 were conducted
in 6-in.-and 14-in.-wide jets and on foils No. 5 and 6 in
a 14-in.-wide jet. Steps (a) through (g) were repeated
for each of these runs.
(j). In all the experiments' $0 far discussed the foils were
located on the approximate center line of the jet. As a
final step, in order to investigate the effect of submer
gence, foil No. 2 was operated in a 10-in. wide free jet
with one side of the jet first 3 in. and. then 2 in. from
the cavity side of the foil. Steps (a) through (g) were
repeated for these conditions.
7
The various readings obtained during the experiments were punched
on data cards and processed in an electronic digital computer. The computer
stored all calibration information and was programmed to yield the following
parameters:
Independent variables
pQV. C = .1
J cspri-j2 (2)
8
(J =
(applicable to ventilated cavities only)
Dependent variables
c = L L " _2 ; L = (L U + L L)
CSpTr/2
, cos a .. D
, t sin a
c = D D csptf/2
, , " , D = D cos a + (L U + L L) sin a
M "" 2 2 ; M = 0.07(L U - L L) - meL U + L L) c spu /2
ACM = CM .. CM
C. = 0 J
Xr. Xr. X:r. A-=---c c c C. = 0 J
(4)
(5)
(6)
(8)
(10)
(11)
(12)
9
M is measured in foot pounds when L is in pounds and is measured about the
mid-chord of the foil (positive nose up), m is the distance in feet between
the mid-chord of the foil and the geometrical center of the dynamometer given
in Table I, XI, is the location of the center of
mid-chord (positive toward the leading edge), and , ,
pressure with respect to , a is the measured angle
between the D -axis of the dynamometer and the vertical.
Other independent variables recorded directly were angle of attack , a( a = a - half wedge angle), jet flap angle ,. , and air-flow ra te Q A
(ventilated cavities only). The cavity length 1, was a dependent variable
recorded directly. Neither QA nor 1, are presented in the data which fol
low.
Much of the data appear in the photographs of Figs. 5, 6, and 7 and
in the graphs of Figs. 8 to 26. Figures 8 to 11 give CL, CD' CM' and
~/c for . Cj = 0 as a function of 0, mainly, but also of jet width and
chord length for two-dimensional foils. Similar data for finite aspect ratio
foils are contained in Figs. 19 to 21. The CD data in Figs. 9 and 20 (and
also in Fig. 12) have been corrected by subtracting calculated skin friction
drag of the end disks and of the test plate so that only form drag appears
[5J. Any form drag on the end disks and wi thin the dynamometer cups was
assumed to he negligibly small. The correction was 0.011 for the 2-in. chord
foils and 0.009 for the 2.75·-in. chord foils at 40 ~ 4 fps. No other cor
rections have been made to the data.
Figure 12 shows typical coefficients for C. > O. Few of the data J
are presented in this form, however, because increments in the coefficients
as defined by Eqs. (9) through (12) show the effects of the jet flap much
more clearly than do the coefficients themselves.
It should be noted that the coefficients as defined in the text are
the measured coefficients, including the effect of the jet flap_ Thus, the
measured lift and drag coefficients include both the part of the lift and
drag contributed by integrating the pressure over the exterior surface of the
foil and the part contributed by integrating over the internal ducting system,
the latter being equivalent to the horizontal component of the jet reaction
in the lift direction and to the tb"rust in the drag direction. The true form
10
drag coefficient with the jet flap operating can be obtained by adding the
thrust coefficient to the measured drag coefficient (corrected for skin fric
tion) and this is the quantity which it is· attempted to plot in Fig. 12.
The other coefficients in Fig. 12 are the measured coefficients, however, and
include the effects of the jet flap. (It might also be noted in passing that
had the jet flap fluid been extracted from the main stream instead of being
supplied from outside the tunnel, thethr~st correction in Fig. 12 would have
been reduced by the mo:nentum of the extracted fluid.)
Figures 13 to 17 contain all of the data on the increments bCL-
two figures, one for cr~O, and one for various cr, bCD' bCM, and bli/c
as functions of C., mainly, but also of cr, jet width, and chord length J
for the two-dimensional case; Figs. 22 to 24 contain similar data for the
finite aspect ratio case; and the data for small submergence are contained
in Fig. 18. The actual values of CL, CD' CM' and lifc with Cj f 0 other
than those shown in Fig. 12 ma.y be reconstituted by adding the incremental
values to the values of the coefficients at C. = O. J
IV. SOME OBSERVATIONS OF THE FLOW
Figure S illustrates the jet flap in super cavitating flow. The
issuance of the flap fluid at the trailing edge of the foil is clearly apparent
in Fig. Sb. The white appearance of the flap at issuance is caused by the
outgassing of the pressurized city water used for the flap as it emerges into
the low-pressure tunnel flow. Water vapor and air bubbles from the cavity
are also entrapped in the flap fluid as it nears the trailing end of the
cavity.
Ho's theory [3J predicts an increase in cavity width near the foil
with C. at cr = 0; his Fig. 9 shows a cavity at J
a = 0 degrees, C. = 0.1. J
A split cavity in the tunnel produced with foil No. 4 at a = 8 degrees and
C. ;, 0.1 (cr~O) is qualitatively similar to Ho's figure, but it is impossible J ..
to make a quantitative comparison. The clear part of a split cavity remains
at about the same size with and without the jet flap so that any apparent in
crease in size is associated with the milky-looking fluid which abuts on the
clear cavity space. Itis difficult to judge whether the milky fluid is part
of the flap fluid or part of the cavity.
11
For u> 0 the cavity dimensions in Fig. 5 appear to have increased
without a comparable decrease in cavitation number when the flap is operating.
However, the entire left half of what appears to be cavity in Fig. 5b is a
gas-liquid mixture, as is most of the lower part of the cavity. It is very
difficult to delineate the true cavity. Cavity pressure and cavitation num
ber remain nearly c.onstant when the jet flap is introduced, as may be seen
from the data on Figs. 5, 6, and 7 and in Table IIa of the next section. The
gas volume in the cavity is probably not much different than without the flap.
It is believed that the increase in size is largely illusory--that it is ass
ociated with the numerous gas bubbles entrained in the flap fluid.
Figures 6 and 7 illustrate the effect of the jet flap on a venti
lated cavity and also show the comparison between two-dimensional and finite
span jet_flapped hydrofoils. (The latter are more photogenic because of the
absence of the front dynamometer unit.) Ventilation of a cavity without jet
flap reduces the cavitation number, increases cavity length, and decreases
lift and drag (compare a and c of each figure). The jet flap increases
the cavitation number and the lift by scrubbing out the air (compare c and
d of each figure), but a similar effect could be achieved simply by cutting
off the ventilation. Hence, it does not appear that a jet flap will be very
useful with deliberately ventilated, supercavitating hydrofoils. At larger
angles of attack than those used in obtaining Fig. 6 and 7, a jet flap with
limited C. is not quite so effective in scrubbing out all of the ventilated J
air; this may be seen in Table II of the next section.
Observation of the experiments showed that if the reference cavit
ation number for a ventilated cavity, a , was so large that a cavity existed v only by virtue of ventilation, then addition of a jet flap caused the cavity
to disappear. (This observation was true for the largest ventilation rates
obtainable in the free-jet tunnel but may not be true at larger rates.) It
does seem therefore, that when cavities are inadvertently produced by ventil
ation from the free surface through a strut or by trailing edge vortices, the
jet flap may be useful as a device for removing some of the air and thereby
increasing the cavitation number. This use of a jet flap would increase the
lift even more than would the jet flap alone in super cavitating flow with
constant cavitation number.
12
There is no fundamental difference between two-dimensional and
fini te-span hydrofoils that may be seen in Figs. 6 and 7. Differences are in
quantity only, and the quantitative results are discussed in the next section.
Particularly apparent was the absence of any startling change in the trailing
vortex pattern of the finite span foils when the jet flap was suddenly turned
on at maximum rate or shut off. (Trailing vortices, if they exist at all in
this tunnel, trail from near the outboard end of the cavity rather than from
the outboard end of the foil itself as may be seen in Fig. 7.)
One other observation pertains to flows with the cavity side of the
hydrofoil within one chord of a free surface in the tunnel. Under these con
diti-ans it was difficult to maintain a cavity of moderate length without
splitting (opening to the free surface). The addition of the jet flap made
it still more difficult, requiring that all of the cavities near the free
surface be observed either at very high 0' (very short cavities) or near
zero cavitation number with no data in between.
v. DISCUSSION OF NUMERICAL DATA
A. Two-dimensional Experiments, C. = 0 J
The C j = 0 data plotted in Figs. 8 to 11 serve two purposes. First,
the increments in lift, drag, moment, and center of lift attributable to the
jet flap are obtained by subtracting the measured values at C j = 0 for each
run (without skin-friction correction in the case of CD) from the measured
data with C. f O. And, second, they serve to check the dynamometer and ex-J
perimental installation as a whole because these measurements may be compared
with earlier measurements obtained with another dynamometer and with available
theory. In the figures are plotted some exact theoretical results for flow
past a flat plate in a free jet at 0' = 0 [5J. Also shown are approximate
theoretical results in infinite fluid at other 0' from WU [6J. Reference
to [5J will also show the comparison with previous experimental data. On the
basis of these comparisons it is believed that the dynamometer and test in
stallation were functioning reasonably well during the experiments.
In fully cavitated flows over a flat plate all forces other than
friction act normally to the plate surface. (This contrasts with the wetted
13
case where a singularity must be introduced at the leading edge of a plate.)
Thus, for the fully cavitated plate with C. = 0, J
(13)
where
with
have
Cf c. =
J had to
is the estima. ted skin friction correction. Analysis of the data
o ·indicated that in order to satisfy this equation, tan (J., would
lie between about 6 and 9 degrees for the 8 degree angle of attack
data, 8 and 13 degrees for the 11 degree angle of attack data, and 18 and 22
degrees for the 20 degree angle of attack data. The computed angle was con
stant for each group of runs associated with one setting of the angle of a
ttack. Thediscrepancies may lie in the estimation of the skin friction cor
rection or in neglecting pressure effects on the edges of the end disks sup
porting the foils and in the dynamometer cup; or, there may actually be var
iations in flow direction of the tunnel stream when a body is under test,
although the near constancy of a, over a range of (J--and hence CL--seems
to preclude this. Henceforth, the test angles are referred to by their nom-
inal values, 8, 11, and 20 degrees, even though the actual. values may be dif-
ferent.
-There are two differences between tests in the free-jet tunnel and
in infinite fluid. First, the free jet boundaries impose a different stream
line pattern and pressure distribution on the body in the tunnel than in in
finite fluid. Figures 8 a~d 9 show that the difference is appreciable near
cr = 0, but that it is less important at larger (J and is probably insigni
ficant for (J > 0.1 for the jet width-chord length ratios used in these ex
periments. As already noted, most of the present experiments were conducted
ina 10 ... in. wide free jet. Two different chord lengths were used in the lO
in. jet and some experiments were also conducted in 6_in. and l4-in. wide
jets; these show the expected variation with chord length-jet width ratio
near (J = 0 and little variation at higher (J. At (J;::: 0.1, the free jet
results can probably be translated directly to infinite fluid results with
little error.·
The second difference is produced by the fact that the free jet is
deflected by a lifting body so that the velocity at downstream infinity is in
a different direction than that at upstream infinity. The upstream velocity
14
has been used for reference in all computations herein and it has been assumed
that the upstream veiocity is in the vertical direction as it is in the empty
tunnel. The effect of jet deflection is analyzed briefly in Appendix C. Al
though there is an induced drag associated with the jet deflection, no attempt
has been made to take this into account in reducing the data since the form
drag, itself, is also changed when the deflection is altered by the lift. Had
measured drag coefficients been corrected for induced drag effects, the angle
of attack indicated byEq. (13) would have been even further below the measured
angle. (Itisof incidental interest that the effect of lift on jet deflection
is apparent in the photographs of Figs. 5, 6, and 7. Comparing Figs. 5a and
b, for example, the right edge of the free jet, marked by the junction of clear
and foaming regions on the right of each photoJ6raph, is deflected a little
farther to the left in Fig. .5b where the lift is augmented by the jet flap.
This swinging of the free jet from side to side as the jet flap is turned on
and off is readily apparent to an observer standing at the tunnel.)
B. Two-dimensional Experiments, C. > 0 J
The data for C. > 0 shown in Fig. 12 look reasonably smooth and J
show no more scatter than those for C. = O. These data are typical of all J '
of the data for C. > 0 at the plotting scales used in Fig. 12. J
Calculated points from Ho's theory at cr = 0 [3J are also plotted
in Fig. 12. It is seen that ,for lift coefficient the theory does not agree
with the data at cr = 0 h'J.t does agree with data taken at cr = 0.21. Ho' s
calculated drag values agree better with the experiments. As already men
tioned, in the drag graph CT has been added to the measured drag coefficient
(corrected for skin friction) to obtain form drag, which is what was calcu
la ted by Ho. Actually, it has been assumed that CT = C j in accordance with
Appendix C in plotting the drag data in Fig. 12; this assumption will be dis
cussed further in Part C, following. It was not possible to reduce Ho's com-
putation for moment to reasonable values at small
points are shown in the other graphs of Fig. 12.
C.; hence, no theoretical J
As already mentioned, the increments in the coefficients show the
effects of the jet flap better than do the coefficients themselves. However,
the plotting scales are exaggerated when increments are plotted and this makes
15
the data appear to scatter more. Figure 1) contains some of the most important
results of the investigation. It shows the measured increment in lift coef
ficient as a function of C. for the split cavities (0- '" 0). (It should . J
be noted that in this and the following figures, the data points with slashes
indicate tests in 6-in. or 14-in.-wide, free jets as opposed to the bulk of
the tests conducted in the 10-in.-wide jet.) On the same figure are plotted
the calculated points based on Ho's linear theory [)] for 0- = 0 as well as
parabolic lines of the form of Eq. (1) drawn as nearly as possible to fit the
calculated points. The equation of these lines is
ACL = 2.10 ~ sin T (14)
There is ,very little variation in ACL with a' by Ho's theory.
, 0.,
The experimental points are seen to fall well below the calculated
points (which are for infinite fluid). There is no ready explanation for the
discrepancy. It does not seem likely that it is associated with the differ
ence between the free jet and infinite fluid boundary conditions since change
in jet width and chord length had no apparent effect on the data as may be
seen in the figure. Furthermore, by plotting increments as has been done in
the figure, the effect of boundary conditions on the basic flow should be at
least partially removed. The possibility that C. has been incorrectly ca1-J
cu1ated must be considered. If cS in Eq. (2) has been underestimated by
measuring it in the dry tunnel, C. could be smaller. That C. can be suf-J J
ficient1y small to account for the discrepancies in Fig. 1) seems unlikely,
however; first, because C. would have to change by different amounts at each '. J
T and, second, because CT > C. at small a if C. becomes materially J J
smaller, and this should be impossible. At small T' and a the data ap-
proach more nearly to the theory than at larger values and, perhaps, if it
were possible to make experiments at still smaller T and a, the theory
might be verified.
Empirical parabolas have been drawn to represent the ACL data· at
the smaller angles of attack for each flap angle. Fortunately, these parabo
las for T::: 60 degrees, 50 degrees (not plotted in this report) and )6 or
)8 degrees all fit the single equati~n
16
(15)
For T = 20 degrees the coefficient is larger, 1.58. The choice of a parabo
la as reference curve is purely arbitrary, of course; clearly, a more linear
variation of ACL with Cj would fit the data better. (The more linear vari
ation at small C. is also true of experimental data for fully wetted foils.) J
Data for ACL versus C j for various 0', including the data from
Fig. 13, have been plotted in Fig. 14. The empirical parabolas from Fig. 13 have also been plotted for referenceo It is seen that 0' has only a small
influence on ACL compared to the influence of 0' on CL in Fig. 12. There
is a tendency for ACL to decrease a little with increasing 0', as the cavity
shortens. (Of course, when 0' becomes so large that the cavity disappears,
ACL would be expected to increase suddenly to its value in fully wetted flow;
the foils in these experiments were not designed for operation in fully wetted
flow so that the occurrence of the discontinuity could be verified.)
Increments in drag coefficient are plotted in Fig. 15. Calculated
points from Hots theory [3J are also shown. (CT = Cj has been subtracted
from Ho's calculated CD values so that the jet thrust is included with the
drag in these figures.) The correspondence between theory and measurement is
surprisingly good at the smaller 0' and T values. The comparison appears
to substantiate the theoretical prediction that the jet momentum is fully re
coverable as thrust even in the supercavi ta ting case if the flap angle is less
than 60 degrees, but this point will be discussed further in Part C, following,
Surprisingly, the measured drag is much too small compared to theory at the 20
degree angle of attack. To find the true increment in form drag coefficient
associated with the jet flap, the ordinate of each data point in Fig. 15 should
be increased by CT (~ C j) • If this were done there would be a slight in
crease in CD with Cj at all 0' as in Fig. 12.
Increments inmoment coefficient and in center of pressure, plotted
in Figs. 16 and 17, show the expected effect of the jet flap, movement of the
center of lift to the rear, even past mid-chord, and reduction of the nose-up
moment about mid-chord. In fact, the trend in increment in moment coefficient
17
C. and in center of lift is linear with J
Approximately and empirically,
and apparently unaffected by angle
of attack.
~CM ~ .. 0.32 C. at T = 20 deg J
(16a)
~CM ~ - 0.50 C. at T == 36 deg J
(16b)
~CM ~ .. 0.56 C. at T = 60 deg J
(16c)
Similar equations could be written for ~Yi/c. If Eqs. (15) and (16) are com
bined, it is seen that .. ACm is proportional to v'C. at fixed T. This ~C J
L is the result from fully wetted flow, as may be seen in Fig. 10 of [2J,:£or
example.
C. Thrust Recovery
With C. > 0, the lift due to pressure on the foil exterior is aug-J.
mented by the component of the jet reaction in the lift direction, as already
noted. Similarly, the drag is reduced by the thrust and the latter has so far
been taken equal to the jet momentum in accordance with the discussion in Ap
pendix C.
Two possibilities now need to be examined. First, C. as measured J
may be incorrect because of an error in estimating /); then the true
fC. where f has to be evaluated. Second, CT ma.y be less than C. J J
C. is J
so that
CT == flCj where fl<l and has to be evaluated. Either f or fl may be
evaluated by making use of the fact already used in Eq. (13) that the external
form drag on the fully cavitated plate is equal to the lift due to pressure on
the exterior of the foil times the tangent of the angle of attack. Thus, if
the true C. J
is fC. , J
(17a)
18
If
(17b)
The factor f from Eq. (17a) has been computed for a number of runs
and the results are plotted in Fig. 25. In the computations, a has been
taken as the angle determined from Eq. (13) for C. = 0 for each run rather J
than as the nominal angle of attack. The data scatter considerably. (Many
factors contribute to the scatter; one of the more important ones may be the
production of forces on the end disks by operation of the jet flap.) In spite
of the scatter, it is apparent that f varies with both T and a but not
with chord length or change of foil; this appears to indicate that C. and J
o are correctly measured and that f should be approximately unity.
Now, from Eq. (17b),
fl = f - (1 - f) tan a sin (a + T)
The values of. fl have been computed from the f-va1ues in Fig. 25, but are
not plotted in this report; f1 is closely equal to f for f near unity
and is only slightly smaller for small f. Referring again to Fig. 25 and
reading the ordinate as fl instead of f, it is apparent that there is def
initely less thrust recovery for T ~ 50 degrees than for T::; 38 degrees.
Thrust recovery also appears to decrease as angle of attack increases, and to
decrease a little as flap angle increases even for small flap angle for these
super cavitating foils. The actual thrust recovery for these super cavitating
foils appears to lie between C. and C. cos (a + T); this is definitely less . J J
than for the fully wetted case. These conclusions would be modified somewhat
by adjustments betw!3en f and f 1 ; but it is believed that they give a true
qualitative picture of thrust recovery in fully-cavitated jet-flapped hydro
foils.
D. Ventilated Cavities with Jet Flap
The effect of the jet flap on a ventilated cavity has already been
mentioned in connection with the discussion of flow observations. Table II
19
contains some typical numerical data for CL and ~CL (~CD and ~CM show
similar t~ends). Beginning with a non-ventilated cavity on line one of each
run. the data show that ventilation decreas'es (J and CL' line two. Then,
the addition of the jet flap, line three, increases both (J (by washing out
most qfthe air as previously obser:ed) and CL• The increase in CL is a
bout equal to the sum of the increments independently attributable to increase
in (J and to Cj • This is shown in Col. (6) of Table II where ~CL due to
the flap alone is indicated; ~CL may be compared with the measured value
without ventilation in line four of the table or with the graphical data in
Fig. 14.
Comparison of the (J values in lines three and four of each run,
Col. (2) of Table II, shows that, at angles of .attack of 8 and 11 degrees, (J
and, hence, cavity pressure are about the same whether or not the cavity is
ventilated when the jet flap is operating. However, at 20 degree angle of
attack the jet flap does not "scrub out" all of the air, and the ventilated
cavity does have a smaller (J. and larger cavity pressure even though the jet
f1a p is operating. Presumably, if larger C j were available, more air could
be scrubbed out. Thedata in Table lIb indicate a substantial decrease in (J
with jet flap and without ventilation when lines one and four of each set of
data are compared. Decreases of several thousandths in (J \uth activation
of the jet flap had been observed in. experiments at smaller C . , (see Table J
IIa), and were considered unimportant; the experiments leading to Table lIb
were the only experim~nts showing such a large decrea se, but time did not per
mit re-running them after the computations were completed. Whether changes in
tunnel flow were occurring, whether these are true effects of large C., or J
whether there were mechanical errors in data taking is not known.
20
TABLE II
VENTILATED, JET-FLAPPED HYDROFOILS (Tested in 10-in.-Wide Free-Jet Tunnel)
a. Foil No. 2
(1) (2) (3) (4) (5) ( 6) (7) Run a, Deg. cr cr c. CL 6CL Remarks v J
0.169 0.169 0.000 00288 Not ventilated
1 8 0.048 0.169 0.000 0.197 Ventilated 0.165 0.169 0.087 0.487 0.199 Ventilated 00173 0.169 00087 0.506 0.218 Not ventilated
1«';-' ".
0.112 0.112 0.000 0.250 :Not ventilated
2 8 0.050 0.112 0.000 0.186 Ventilated 0'.105 0.112 0.083 0.450 0.200 Ventilated 0.109 0.112 0. 083 0.458 0.208 Not ventilated
0.233 0.233 0.000 0.393 Not ventilated
3 11 0.089 0.233 0.000 0.260 Ventilated 0.220 0.233 0.091 0.576 0.183 Ventilated 0.242 0.233 0.091 0.605 0.212 Not ventilated
0.146 0.146 0.000 00323 Not ventilated
4 II 0.067 0.146 0.000 0.259 Ventilated 0.134 0.146 0.085 0.508 0.185 Ventilated 0.138 0.146 00085 0.524 0.201 Not ventilated
0.438 00438 OgOOO 0.723 Not ventilated
5 20 0.224 0.438 0.000 0.508 - * Ventilated
0.295 0.438 0.103 00737 0.172 Ventilated 0.437 0.438 Ogl03 0.892 0.169 Not ventilated
0·314 0.314 0.000 0.579 Not ventilated
6 20 0.161 0.314 0.000 0.462 *
Ventilated 0.227 0.314 0.094 0.692 0.172 Ventilated 0.296 0.314 0.094 0.779 0.200 Not ventilated
0.224 0.224 0.000 0.531 Not ventilated
7 20 0.153 0.,224 0.000 0.474 - * Ventilated 0.182 0.224 0.089 0.675 0.182 Ventilated 0.215 0.224 0.089 0.721 0.190 Not ventilated
21
,
TABLE II
VENTILATED, JET-FLAPPED HYDROFoILS (Tested in 10-in.-Wide ~ee-Jet Tunnel)
b. Foil No. 3
(1) (2) (J) (4) (5) (6) (7) Run a., Deg. er er v C. CL J
ACL Remarks
0.272 0.272 0.000 0.410 Not ventilated 8 11 0.117 0.272 0.000 0.253 - * Ventilated
0.233 0.272 0.130 0.666 0.266 Ventilated
0.434 0.434 0.000 0.667 Not ventilated
9 20 0.210 0.434 0.000 0.460 - * Ventilated 0.303 . 0.434 0.146 0.753 0.181* Ventilated 0·391 0·391 0.146 0.818 0.187 Not ventilated
0.264 0.264 o.oob 0.567 Not venti4ted 10 20 0.142 0.264 0.000 0.454 * Ventilated
0.185 0.264 0.140 0·717 0.232* Ventilated 0.230 0.230 0.140 0.737 0.215 Not ventilated
* A~ is obtained by subtracting from the measured value of CIL an aver-age value of S. at C. = 0 obtained. from Fig. 8 using the measU):'ea. value of cr shown in Co~ (2). J
22
E. Effect of Submergence
As indicated in Section III (j), several runs were obtained in the
lO-in.-wide free jet with the cavity side of the foil within first 1-1/2 and
then 1 chord of one free surface. However, only quite high cr or cr near
zero could be obtained because of the tendency of the cavity to split easily
as noted in Section IV. Da ta for C. = 0 corresponding to Figs. 8 to 11 are J
not presented herein. These data, when compared to Figs. 8 to 11, are quite
similar at high cr but give slightly higher CL and CD and slightly lower
eM near cr = O. These tendencies near cr = 0 are in accordance with theory
for cr = 0 [5J.
Figure 18 shows the increments in lift, drag, and moment coeffici
ents for these data obtained in the usual manner. The lift increments are to
be compared with the empirical parabolas and data in Fig. 14b for the symmet
rically placed foils. The split cavity data at 8-and ll-degree angles of at
tack compare favorably with the symmetrical data, but at 20 degree angle of
attack the lift increment is far too small. The jet flap appears to become
relatively less effective at small cavitation numbers and large angle of at
tack as the free surface is approached. At high cr (small cavities) the lift
coefficient data scatter ~onsiderably; the low data on the ll-and-20-degree
angle-of-attack graphs are commensurate with the symmetrical data. The high
data at 8-degree angle of attack may be in error, but there was no opportunity
to re-run these experiments.
The data for increment in drag coefficient and for increment in
moment coefficient compare well with the data for the symmetrically placed
foil at 8-degree angle of attack and are not too different at 11 degrees , al
though there appears to be a definite influence of depth atone-chord submer
gence on the Il-degree graph. At 20-degree angle of attack these data, like
the lift coefficient data, seem to indicate that the jet flap loses some of
its effectiveness ~ear cr = 0 as the surface is approached.
These data on the effect of submergence are very meager, and the
experiments should be re-run with attention directed strictly to the problem
of submergence. However, on the basis of Fig. 18 it may be concluded that, to
within one chord of a free surface, a two-dimensional super cavitating hydrofoil
with jet flap at small angle of attack and small C. behaves much the same as J
a deeply immersed one.
23
F. Finite Span Experiments
The experimental data on finite-span hydrofoils appear in much the
same form as for the two-dimensional hydrofoils. They are limited to foil
No. 4 with Aspect Ratio of 2.5, as alre~dy indicated, and even for this foil
only a few runs were attempted. Figures 19 to 21 give the basic data at , ,
Cj = 0 and serve the same purposes as Figs. 8 to 11 for the two-dimensional
data. The lift and drag coefficients are compared with Cumberbatch's theory
[7J for infinite fluid, and the comparison appears reasonable.
Increments in lift coefficient, drag coefficient, and moment coef
ficient sholm in Figs. 22 to 24 are not, greatly different in form from the
two-dimensional case. As must be expected, the increment in lift coefficient
is less than for the two-dimensional case. Fitting an empirical parabola to
the data, k in E~. (1) is of the order of 1.10 as compared to 1.27 for the
two-dimensional foils with similar jet-flap angles. In the fully wetted case,
the variation of k with aspect ratio would be expected to be given by the
factor [2J.
2C. 1 + .J
TTm.
1 + ~ + 0.604 At mvC:"
J
+ ---
and this is approximately equal to 0.555 for the conditions under which foil
No. 4 was operated. For the experimental data this factor is 1.10/1.27 = 0.865. This discrepancy is not entirely unexpected in view of the observation
in Section IV that there was no outstanding change in trailing vorticity with
changes in Cj , and that the trailing vorticity, if it was visible,appeared
to originate at the end of the cavity rather than immediately behind the foil'
as in the wetted case. Hence, the loss in lift attributable to finite span
is not nearly so great in the fully cavitated case a.s it is in the fully wet
ted case. ~
The increment in drag in Fig. 23 appears to be slightly more nega-
tive than in the two-dimensional case and this is difficult to explain. Fig
ure 25d shows the thrust recovery factor evaluated as in Part C for the two-
24
dimensional foils. This figure should be compared with Figo 25b, which is for
two-dimensional flow at approximately the same flap angle. Thedata are very
similar, indicating that the thrust recovery is about the same in the two
cases. The comparison further
ated with an increase in Cj if there were, the factor f
suggests that ~here is no induced drag associ ...
for finite-span, jet flapped hydrofoils; for,
in Fig. 25d woulo. have to exceed unity and it
is hardly likely that this is possible. Again, the absence of induced drag
is consistent with the already mentioned trailing vortex pattern.
The effects on moment and center of pressure of changes in C. in J
the finite-span case are qualitatively similar to those in the two-dimensional
';,':L case,--but the changes are less rapid. Thus, although the relation between
CM and Cj appears to be still roughly linear, the proportionality now is
about -0.40 or less instead of -0.50 as given in Eq. (16b) for approximately
the same flap angle.
VI. SUMMARY
There are many similarities between the actio.n of a jet flap in
fully cavitated flow and in fully wetted flow. The differences are largely
quantitative rather than qualttative, but there are exceptions where the dif
ferences may be more fundamental. Also, some problems, such as those associ
ated with ventilation, are peculiar to the fully cavitated flows. The detail
ed findings of this experimental research are summarized in the following par
agraphs. It should be remembered that the maximum jet momentum coefficient
was of the order 0.1 for these experiments.
The lift of a two-dimensional, super cavitating hydrofoil is increased
by use of a jet flap apprOximately in accordance with the formula
(1)
For cavitation numbers near zero, k:::::: 1.27 at jet flap angle ,. = 36, 38,
50, and 60 degrees and angle of attack a. of 8 or 11 degrees. At ,. = 20
degrees, k ~ 1.58 under similar conditions. The coefficient k is smaller
at a. = 20 degrees; k decreases slightly with increase in cavitation number.
2.5
From very limited data for an aspect ratio 2 • .5 hydrofoil, k ~ 1.10; this is
much closer to the two-dimensional value of 1.27 than would be the case in
fully wetted flow. The differenqe from the wetted flow is probably a funda
mental one associated with the much weaker trailing vortex system in fully
cavitated hydrofoils.
The experimentally determined increments in lift coefficient fall
far below the theoretically predicted ones as calculated by Ho [3J for zero
cavitation number where k ~ 2.10. Since the experimental values for small
T approach more closely to this theoretical value and since the theory is a
linear one, it may be that experimental verification should only be expected
at vanishingly small T and a.
Form drag of a super cavitating hydrofoil is also increased by opera
tion of a jet flapo The measured increment in drag was approximately in
accord with Ho's linear theory.
For the supercavitating, flat-plate hydrofoil, lift and drag are
related by
(i7b)
Here Cf is a correction for skin friction, CD includes both form drag and
thrust, and fl is a thrust recovery factor. In the experiments, fl was
found to lie between unity and cos (a + T) and to decrease somewhat with in
creasing T and a. This differs from the wetted case where fl is close to
unity for flap angles up to near 60 degrees. For finite span foils, fl was
found to have about the same values as for two-dimensional foils without mak
ing any allowance fol' induced drag. Again, this differs fundamentally fr??l
the wetted' case where. the effect of induced drag would have to be accounted
for; the difference is explained by the much weaker trailing vortex system in
the supercavitating case.
Moment becomes more nose down during operation of a jet flap and
the center of pressure moves to the rear. The. increments in moment coeffic
ient and center of pressure both tend to vary linearly with the jet momentum
coefficient.
26
Some very limited experiments conducted with the cavity side of the
foil nearer a free surface than the other side indicated that for small angle
of attack with submergence of one chord or more, the increments in lift, drag,
moment, and center of pressure due to a jet flap are not much different than
in the deeply submerged case. However, at 20 degrees angle of attack within
one chord of the surface, the hydrofoil with jet flap showed much smaller ef
fects than in the more deeply submerged case o
The use of a jet flap with deliberately ventilated hydrofoils does
not seem to be useful since the jet flap "scrubs" out the ventilating air.
This increases the lift and drag without question, but the same result could
"::L be acliieved by cutting off the ventilating gas and thereby increasing the
cavitation number. For hydrofoils which are inadvertently ventilated from a
free surface, use of a jet flap may prove very advantageous. In this case,
the flap scrubs out some of the unwanted air and produces a two-fold increase
in lift; one due to the increased lift associated with the larger cavitation
number and the other due to the normal increase in lift produced by a jet
flap at a given cavitation number.
Since the jet momentum coefficient in these experiments was limited
to values of the order of 0.1, the results are most directly applicable to
jet flaps used for producing small changes in lift for trimming purposes. If
the results are to be extrapolated to larger momentum coefficients, this
should be done with caution; the results are possibly extrapolatable qualita
tively, but certainly not quantitatively. The rearward movement of the center
of pressure at moderate jet momentum coefficients is useful in supercavitat
ing flow because the thick foil sections occur far back on the foil. With
high jet momentum coefficients, there may be engineering design problems as
sociated with the high nose down moments and rearward shift of the center of
pressure of the hydrofoil; - ~~/ ~CL is proportional to v'Cj within the
range of these tests, and since this proportionality also exists for fully
wetted foils at higher C. it may also hold for the fully cavitated foils J
at higher C •• J
Problems requiring further investigation include:
Extension of the two-dimensional theory to include
non-linear effects; or, at least, a further inves
tigation of the discrepancy between the present
results and the linear theory.
Detailed investigation of the utility of using jet flaps to control unwanted ventilation.
Further study of both finite aspect ratio effects and the effect of submergence.
Investigation of short, trailing edge, blown flaps. These may offer some advantage in control of the flap angle and wi11 probably increase the lift more than the pure jet flap alone if experience with fully wetted flow can be used as a criterion.
Unsteady flow, both that occurring during starting and stopping of the jet and that associated with the travel of a jet-flapped hydrofoil through waves.
27
28
The work described in this paper was performed under sponsorship by
David Taylor Model Basin of the U. S. Department of the Navy.
K. Yalamanchili and Arthur Pabst were responsible for obtaining the
experimental data and were assistedbyPaul Edstrom and Adiseshappa Rao. The
manuscript was prepared for publication by Judy Mike.
29
[lJ Lachmann, G. V. (Editor). Boundary Layer and Flow Control, Vol. Pergamon Press, New York, 1961.
I
[2J Williams, J., Butler, S. F~ J., and Wood, M. N. The Aerodynamics of Jet Flaps. Aeronautical Research Council, Great Britain, R and M No. 3304, 1963, 32 pages + 21 figures.
[3J Ho, H. T. The Linearized Theory of a Supercavitating Hydrofoil with a Jet· Flap. Hydronautics, Inc., Technical Report No. 119-2, June 1961. 42 pages + 9 figures.
[4J Silberman, E. and Ripken, J~ F. The st. Anthony Falls Hydraulic Laboratory Free-Jet Water Tunnel, University of Minnesota, st. Anthony Falls Hydraulic Laboratory Technical Paper No. 24, Series B, August 1959& 24 pages + 19 figures.
[5J Silberman, E. "Experimental Studies of Supercavitating Flow about Simple Two-Dimensional Bodies in a Jet," Journal of Fluid Mechanics, Vol. 5, 1959, pp. 337-354. ..
[6J Wu, T. Y. A Wake Model :tor Free-Streamline Flow Theory, Part 1. Fully and Partially DeveloRed Wake Flows and Cavity Flows Past an Oblique Flat Ple.teo California Institute of Technology, Report No. 97-2, September 1961. 26 pages + 8 figures, and "A Free Streamline Theory for Two-Dimensional Fully Cavitated Hydrofoils," ~and Physics, Vol. J:J:J::.J, October 1956, pp. 235-265.
[7J Cumberbatch, E. Cavitatin Flow Past aLar California· Institute of Technology, 17 pages + 7 figures.
IlQQR~§'
(1 through 25)
j
32!
3D-in. Dia. Inlet Pipe Test Section Waste Pipe (33 ft high) Jet Width Control Wheel Tunnel Air-Intake Control Reference Pressure Tap Velocity Manometer Pressure Manometer Control Stand
Fig. 1 - The Two-dimensional Free-jet Water Tunnel
33
Fig 0 2 - Front Face of Tunnel
34
a. Dynamo'meter as seen from outside of -tunnel
b. Dynamometer as seen from inside of tunnel
a. Upper beam e. Inlet for ventilation or for
b. Lower beam measuring cavity pressure
c. Tension fibers f. Bellows
d. Inlet for jet-flap fluid g. Dynamometer cup (fits into 33· .
hole in test section wall)
Fig. 3 - The Dynamometer and Two Hydrofoils
a.
b. g.
h.
c. Finite-span hydrofoil mounted in dynamometer
d. Two-dimensional hydrofoil
Upper beam
Lower beam
Dynamometer cup (fits into 3.7-in. hole in test section wall)
End disk
i. Disk cups (fit over beams a and b)
j. Screw for tensioning wire holding finite span foil to dynamometer
k. Paper strip inserted in jet-flap slit
Fig. 3 ... (continued) ... The Dynamometer and Two Hydrofoils
35
If)
o q +II. .
,0 II') (\I
r<i
,End Plat$
,(2 Req'dl.
Foil (I Req'd)
,... 5.06"
Symmetrical about ¢..
V2-
----
to .... ~.5·-" "" 2
<f. ,
(~
(:, ~qJ"
'(\I
"
'Sweat tube to foil
3 fl _1..' '.J:'
~-I_~ 8 0.0. II 4 1.0. brass
, ' ",' sectS A-A tube. ,Tap:.r drill a,fteT
, ass,embly -I'" each erid I" . 'V
to. 4 , on ¢...
All parts fabricated
from brass.
.001
Fig. 4 - Construction Details, Jet-flapped Hydrofoils
c.:i 0-
a. No jet flap it (J" = 0.048 Cj = 0.000
Hydrofoi I no. 41 a =lIdeQ
1/50 sec. exposure
..
b. With jet flap (T = 0.047 Cj=O.IOO
Fig. 5 - Typical Super-cavity without and with a Jet Flap
37
I' ; II·'
I!il
38
a. No jet flap
'" = 0.091 "'v. 0.091
CJ :0.000
'"
b. Jet flap added to(a)
'" s 0.094 .
"'v c 0.094
Cj = 0._0_90 __ -1 .....
c. Air added to (a)
"''' o.oae
... "'v" 0.091
CJ "0.000
d. Jet flap added to (c)
'" =0.083
"'v =0.094
CJ = 0.090
------
Exposure = 15 microseconds
(CX "II deo. T" 36 deg )
Fig. 6 - Cavities Produced by a Two-dimensional, Jet-flapped Hydrofoil
a b
c d
15 microsecond exposure
a. rT = 0..0.73, av = 0..0.73, Cj = 0..0.0.0.
c. rT = 0.0.15, rTv -=0..0.73, Cj = 0..0.00.
(a = II deg
a b
c d
1/50. second exposure
b. rT =0..0.68, rTv =D.D68, C(D.D98
d. rT = 0..0.68, rTv = 0..0.73, Cj=D.D98
T = 33deg)
Figo 7 - Cavities Produced by a Finite-span, Jet-flapped Hydrofoil
39
40
0.6~----~-------r------.-------r------.-------r-
0.5J----+----+----+---F---t-----t-
0.4~-------+--------+---~~~--------~----
[J
,~ 0.31-=-=-=----+---=-~yK.:7~-__:_+--
-o 0.1'--___ 1..
-c Q)
o --Q)
o U
o
0.4
0.2 0
o
0.1
a = 20°
Free jet width
6" 10" 1411
0.2 0.3 Cavitation nurn ber, (j
60°
P 0
'0.
0.4
r: fr hord
2' 36° 20°
I::. 0
0.5
2.7511 Theor. 38° [5]
<"111111 0 <---~ <---
0.6
Fig. 8 - Lift Coefficient as a Function of Cavitation Number, C. = 0, AA. = 00 I
0 u
.. C' C ~
"0 -0
+0-
C Cl)
(,) --Cl)
0 u
41
0.15r-------,-------r----..,-----r------,-------r--------.
0.1 Ol------t-----+-----t----t---~~'------+------'
o
0 o
o
r chord
Freejet 2" 2.75 11 Theor. width 60° 36° 20° 38° [53 0.1 0 t-----+----t-------\ -<11111111
o -<--<----
0.050 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Cavitation number, (j
Figo 9 - Drag Coefficient as a Function of Cavitation Number, C. = 0, AR = 0) I
42
0.15 T chord
Free jet 211 2.7511 Theor width 60° 36° 20° 38° [,5]
0.10 6" P lO" 0 t:. 0 0 ~-
14" Q. ~
r..
i~~ B~& ~ pu
a= 8°
0.05
o
~O.IOr-------+-------+-------~------~-------r-------+------~
~ ~-Wu~ E o E - 0.05·~~d'i5$~~~~~~-4---+---t------l o
-c (1)
0 - 0 -CI)
0 (J
0.10
0.0 oJ
a=W
0 Q)o u..
~ tJetA o 0 ~ 0 \ ¢tJ ~ 0 '¢. --
b'@;P. '¢. t:I§J OIgl ti
Wu v
a=20°
0.1 0,2 0.3 0.4 0.5 0.6 Cavitation number, (]'
Fig. 10 - Moment Coefficient as a Function of Cavitation Number, C. = 0, AA= (X)
I
0.7
0.3r-----~~----~------~----------------;T-------------, chord
211 . 2.75" Free jet width . 60°' 36° 20° 38°'
[J ~
Theor. [5]
0.1 r------+------~--~~r_~--_r------._----~------~
0=80
43
>ilo OL-----~------~----~------~------~----~------~
~ Q) -c Q)
U
,~ O.~~
0.1
o
0.2
O. I
:v.. ~]: ~~ ~~; ~
%. AO A~ l:l A - \
a =1\0
':f o ~~
/!AD 0 ~D~ 0>00. ~ <o~ ~~~ 0 0 0
~ ~D A A L:l V"",,
<> A (1= 20°
0.1 0.2 0.3 0.4 0.5 0.6 Cavitation number, (J'
Fig. 11 - Center of Pressure as a Function of Cavitation Number, C. = 0, AR = 00 I
0.7
44
<.::t.'
~f(,)
CD ... ~
::: 0.2 CD ... Il. -o 0 ... .!!! c: CD
.1
U 0.0
)
-0.1 o
Z U
... c CD E 0 ~ -0 0.10 ... c .! (,)
:: 0.05 Q) 0 iD u
0.00
-o.()~ O·
u~
+ 0 u
'E CD
'u :;:: -CD 0 U -<II ~ ...
.J::. I-III
E Il. -c
CD
u :;: -CD 0
U
c:JI c ... 0
0.10
0.05 +
A·
0.6
.J 0.5 u
--::J 0.4 -0 -C ,~
CD 'u 0.3 :;: - [
CD 0 U
0.2
I
0.02
I:::.
+ POI:::. 0
I:::. I:::. I:::. 0
0 A -=
0 p + 0 p
(D V <p
P
0.02 0.04 0.06 0.08
Jet Momentum Coefficient. CJ
+
0.04 0.06 0.08 0.10 0.12
Jet Momentum C oeffi cien t. CJ
~Q:h ~I:::. o~ fllll:::.
0.02 0.04 0.06 0.08 0.1 o
Jet Momentum Coefficient. CJ
0.10
RP 01:::. o ~~ ~I:::. 0 ~ o (J 0.004
p"t:::. o (J = 0.12 I:::. (J = 0.21
+ Theory (J =0 [3]
0.02 0.04 0.06 0.08 0.10
Jet Momentum Coefficient, CJ
Figo 12 - Typical Force and Moment Coefficients, c.> 0 I
0.12
0.14
0.50
OAO
0.30
..,,0.20 u <l
.... C:'
.!! o --Q) o 00.10 +-c:
+-c: Q)
E
./ ! // !
I
" I. I ,
II AE'U41 ,,'
," T=60°
.cf ~.4- 1.I0;C:;
V P" ~
peJ,,-<, r' I e..
I ,..., e.. ~ " " ~
, ~: b fII(
/ ,~' , ,~ ~ e
.d ~ tlJ e ; I/, '<D
/fP. ~ e.. , e ;L'~,.(l41 I ~ T=36° - LEGEND-Free Jet width, in.
~ II cx;I 10 6 14 theory a [3]
T,deg C,in. 8" 8° 11° 2Cf'
/ ~ 20 2 o I!'! 4;).,,, • 'iii 0.7 4./C,; 1~ ~" ~ 38 2.75 ~ 0 ~ '0.
V ,,~ "I!( v. ~ 36 2 A £ 8. ~
60 2 • Q o e J:f'Q Q) ... o c:
00 0.02 I .... '" ~~ ~
0.10 f-I--./r-7:--'I=---_+_-
0 0 0.02 0.04
00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Jet momentum coefficient I CJ
Fig. 13 - Increment in Lift Coefficient near cr= 0
45
0.20
a. T = 20° o IT-O
o IT Int-ermediate .t::. IT > 0.20, a =8°
IT > 0.30, a=UO fT > 0.40, 4"20°
.t::.
0.151 I~
oJ u <I
...
0.101 .",,""'OI.~'-'
~ 0 o
~ 0.10
.!:
~ 0.05 t-I ..... ;f-"-+----I
E CD .. o c - 0
0.10 i 741 () le
0.051 / - I
4 = 11°
.t::.
00 0.02 0.04 0.06 0.08 0.10 0.12
Jet momentum coefficient, C"
.t::.
0.14
0.20
0.15
.f
b. T =36· o IT ~ 0 o CI' Intermediate
.t::. CI'> 0.20, 0=8° CI'> 0.30, 0=11· IT> 0.40, 0=20°
'Q. 14 n jet Width".48~.;o;..:Di~'--_-I-__ .J.-_---1
• 2 .. 75"chord ' -foil
..la.lol -;: ~ u ~ Z).r
<I
c 005 1-1 -I-l--+--CD •
o --CD o o
:: 0.10 i A ""'.I~
c
III C 0.051 ! CD
E CD 'o c 0
0.10 I 74 ::£0 ! ;. .,.
0.051! I A .. I
O~I __ --L ____ ~ ____ ~ ____ ~~ ____ L-____ ~ ____ ~
o 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Jet momentum coeffiCient, Cd
Fig. 14 - Increment in Lift Coefficient
8:
47
0.50
C T = 60°
0.45 0 (j -0 0 (j intermediate A (j >0.20, a = 8°
0.40 (j > 0.30, a = 11° (j > 0.40, a = 200
rJ 6" jet width
'Q. 14" jet width 0.35
a = 8°
0.30
0.25
0.20
A ..J 0 U .
<l0.15
-c .!! 010 .~ . 0 0.04 - 0 -Q)
0 0
- 1.I0~ -c
C 0.20 Q)
E Q) ... u -= 0.15 o
o
0.10
0 0
a = 20°
0.15~--~r-----~A-~~----r-----~----~----r-----r-----r---~
0.100~---''-:=-!::-:::----::0:-:!.0:-::4:----::0-!:.0:-=6---=-0.~0-=8--0::-.!-::10:-----::0:-!.1-=2-"'-::0~.14-:----=o.-!-:16:---o::-.~18:-----::0:-::.20 Jet momentum coefficient, CJ
Fig, 14 - (continued) - Increment in Lift Coefficient
~~~~~-~~~~~ =:;:-=-~~~::-=~~~~~-"--"-----==:::-=-=-=--==~---=- ==-------=--'-~-=--- - ------
0.025
o
-0.025
-0.050
o u <l .0-C 0)
o +
c§ ~1 ~, 0
cfu~ ~o ~
0
a, I: = 20°
0 0" ~O
0 0" Intermediate
L::. 0"> 0.20, a=8° 0"> 0.30, a=llo cr> 0.40 a =20°
+ Calculated [3]
I~ L..I 1
€tJL::. a=8°
-+ I I I ~I I I i-o.o25! ! 0 !~'" L ~ --~. a=llo
"0
.!:: -0.050'--.0-C 0)
E 0) ~
<.> c
o
-0.025
-0.050
-0.075 o
+ 10 Ino I·
o
L+-1
0.02 0.04 0.06 0.08 0.10 Jet momentum coefficient, CJ
0.12
0.050
0.025
I + 0
-0.025
0 u <l "i-O.O?O 0)
<.> I + ;;:: -0)
0
Or u
en 0 ~
"0
.!:: -0.025 .0-c 0) + E 0) ~ 0.025 <.> E
0
-0.025
)~
Ii
b. I: "36°
o cr ~ 0 o Intermediate L::. cr> 0.20, a = 8°
cr> 0.30, a = Ifo cr > 0.40 a =20°
"Q 14" Jet Width 2.7S"Chord Foil Calculated [3]
a=8°
a=llo
+
a=20D
L::. L::.
o
002 0.04 0.06 0.08 0.10 Jet momentum coefficient, CJ
-0.050,::"' __ -L-__ -L-__ .l.-__ L-_---1L..-_-1 __ ...J
o 0.12
Figo 15 - Increment in Drag Coefficient
..,.. 00
Q u <l
-c
0.050
0.025
0
-0.025
-0.050
0.050
.~ 0.025 ;;:: -Q)
o o en 0 o .... '0 C
~-0.025 E Q) ... o c
0.100
0.075
0.050
0.025
+
+
+
49
t:f9 ~~ ~~ a =S" .ca I.l/!:S"J:Il.JT ~ <.>
~~ ~ .... £:.( 0 £:. Po £:. /).
+ C T = 60°
0 (I'~O
0 (I' intermediate £:. (I' > 0.20, a = So
(I' > 0.30, a = no (I' > 0.40, a = 200
P 6" jet Width p( PPo§) K~ lei 0 '0. 14" jet width q; ~~ 10
p }J + calculate d [3J 0 <P ( g I :~ ~ 0 , 0 £:. 0
£:.
a=IIO
I I
J + I
a =20° +
Jf P ~ 0 .s::b ~@. 0 ()
P £:.£:. ~ ~? ~ 0 l:.
IP ~
0 0 0
t ~~ "- -..
0.02 0.04 0.06 O.OS 0.10 0.12' 0.14 0.16 0.18 0.20 Jet momentum coefficient. CJ
Fig. 15 - (continued) - Increment in Drag Coefficient
E u
-0.05
-0.04
-0.03
<l -0.02
1: Q)
'0
:: -0.01 Q)
o u
I::. 0
0 OM 0
o p. 00 0
th Jil 0 0 Po n J:
a. T = 20°
0 0" ... 0 0 0" Inte rm e diate
I::. 0"> 0.20, a = ao
0"> 0.30, a=llo
rr> 0040, a=20'
I::. I::. Jao 0
[10
I
-c o 0 E -0.03
E
.!: -0.02
-c ., E f -O.QI u c
-0.03
-0.02
. -0.01 o
a=IIO
0.02 0.04 0.06 0.0 a 0.10 0.12 0.14
Jet momentum coefficient, CJ
~ , r-:
-005r---~------r----r---~~r-------------
. I I 10 1::.1 b. T = 36°
-0.041 i 1 lO 11 g ::~n~ermediate • I::. 0"> 0.20, 0= ao
iIP. rr> 0.30, 0 = 11° .-1::.. . 0'> 0.40, 0=20°
-003 ~. ~ 14" jet wIdth
~ . _ __ .• 2.75" chord foil
a = ao
.:-0.021 I" ~ c CD
u -'; -0.01 o u
~ CW', 0 e-0.03 Ii· o E
c '- -0.02 1 ill --c ., E Q) ~ -0.01 ~I __ --'-__ --1_
.5
-0.03 i ~ G4:.l ~ I
-0.021 ~f.)r) • I::.
I::.
-0.01' .
o
o 0.02 0.04 0.06 0.0 a
o
o
a = I JO
~ A
a = 20°
0.12 0.14
Jet momentum coeffiCient, CJ
0.16
Fig. 16 - Increment in Moment Coefficient
01 o
-.07
-.06
-.0.5
-.04
-.03 :E
0 <I
-.02 ~ -c
Q)
'0 .... -.01 .... Q) 0
0
'E -.04 Q)
E o ~
.= -.03
'E Q)
~ -.02 ... o c
-.01
-.04
-.03
-.02
I
c.
L... ...
51
T = 60·
0 (]" = 0 0 (]" Intermediate
/:::,. (]" >0.4 Ct· 20° /:::,. (]" >0.3 Ct· 11° /:::,. (]" >0.2 Ct- 8° 9f 6" Jet width
a = 8° '&. 16" Jet width
/:::,.
/:::,. /:::,.
o
o ~
0 0
~ L~
/:;
9f /:;
n. O 0
~~ ~ ~ SJ. u
J>C ~~[ F~ ~ 0
() ~ ;o~ ~
~ ¢ 0 Lf' 0
/. A 0 0
~ ~ /:::,. 0 a = 20° 0
0 0 /:; ."-" I'-' w D ~ ~o 0
COO /:; ~ .... ....,
~o'~ u
(Cb p ~ 0 0
~ IClJ ~
~
.02 .04 .06 .08 .10 .12 J4 J6 .I~ .20 Jet Momentum Coefficient, CJ Fig. 16 - (continued) - Increment in Moment Coefficient
52
'"', ""'--
-0.2
I oJl -0. XU
<l CIl ... ~ 0 CIl
~ 0. -0.2 -0 ... Q.) ....
I c: Q.) -0. u c:
.... c: Q.)
E Q) ... 0 .=
o
-0.2
-0.1
-0.3
-0.2
><'10 <l -0.1 Q) ... :l CIl CIl Q.) ... 0. ....
o
I a. T ::.20"
~ ["',-, 11O"\..A.~ t#JA <&C ~~ pu---
(1=8°
rP~ h~ 0
(
ci ~L.~ P!CJ:J~Q). 'i1 l
A (1= 11°
(1= 20°
~t Oq ~~ ibJ tjAQ] J A
& A f1A A A
0.02 0.04 0.06 0.08 0.10 Jet momentum coefficient, CJ
0.12 0.14
b. T::. 36°
lI' .. ~ 0 (j~0
0 (j intermediate
.~( • roi !D 8 (j >0.20, (1::. 8°
~~ (j>Q.30, a=llo
r-.,~ ~ /\... ~ (j >0.40, a=200 h-I~ j5'L.>. 1" ~ .14" jet width ~ .... "'
a=8° • 2.75"chord foil
~ -0.21----1----+---+----+--...----t----+-----1 Q.) -c: Q.)
o
.= -O.II-------Ir---t3fi~~~~n=......,...--=-+----t---1 .... c: Q.)
E ~. o~--~----~----~----~----~----~-
-~ o . c: - -0.21-------1----+----+---+---+---+----'-1
o -O.II---+---t---n--h...-t"1--1lHI1-~---
+----1
'-',.0 A
O~ __ ~~ __ ~~~~~~~ __ ~~~~~~ o 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Jet momentum coeficient, CJ
Figo 17 - Increment in Center of Pressure
-0.4
c. T= 60° -0.3
P P' 0 0" ..... 0 p a =8° 0 0" intermediate A 0" >0.20, a:: 8°
~.nt :> ( °A )
A 0" >0.30, a= 11°
~ lffit °A 0 0 0">0.40, a=200 :o.~ ~ If 6" jet width p-b. 14" jet width
-0.2
-0.1
o
~0.3~~~-----+-----+----~----~----;-----+-----r-----~--~ a=20°
p
0 0 p !XJP p
0 0 0"",- ~ 0
0 ~ 0
-0.2~--~r----+-----+----~--~~--~;-----+-----r-----~--~ o o
~ ~ ..4: m. ~U" u~ ....
°O~--~~--~~~~~~~--~~--~~--~--~~--~~--~ 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Jet momentum coefficient, C;,
Fig_ 17 - (continued) - Increment in Center of PreSsure
53
I i,·:
=-::::'-:~---::-';:::::-;O""",::;:=:::=-=-=-~.
_-;;. _:..... ___ ~_~
------------
0.35. • Id
0.301 I D
0.251 I··
0.201 1"""""'-
..... c .~ O.o5l'>l-• .I.'--I--t-" (.)
;;::: ~ 0 o (.)
..... .... = 0..1 c
"E 0.0 Q)
E Q) .... (.)
c
0.1
·0.0
00 0.04 0.08 0.12
Jet momentum coefficient, CJ
0.16
h
d-Id'[l!!c--,31n. 21n. ~ c:r-O 0 •
°1 + 1 r '">0 ~ cal: 131
~~~: + I~ /9/ al~ / / I
c· .02 Q)
2 ttQ)
8 -.02
~ + I )
5 C> o -e-.OfD )t-
c .- -.07~ 5~ I
I
~
£ •
"E + Q) 0' , ,+ E.2 5 Q) .... Co)
E J
-.02 .;
-.ofD
-.0750
n
0.04
I
I I J 1 a; 11°
o • -0 p • ~.
t • I I • I I
• -.;:;T
o[ • a=20°
~
1 0
0.08 0.12 0.16
Jet momentum coeffici ent. C,r
-0.06
-0.05
-0.04
-0.03
cJ <1-0.02 ~
.~ -om (.)
;;::: 't-0.04 o (.)
1: -0.03 Q)
~ -0.02 E .!: -0.01 ..... c E -0.0 4 Q)
~ -0.0 ., ~
-0.0 2
-0.0 I
I~
Illig I: IJ~
II t 1~:·IO:i"·I· 0 0
[ a =20°
n n Q
-'- -- --- --~ 00 0.12 0.16 0.04 0.08
Jet momentum coefficient, C,
Fig, 18 - Increments in Coefficients for Small Submergence, T == 360 , chord == 2 in.
~
01 .j:..
Q (.)
0.20
Di 0.15 c .. 'C
'0 0 .10 -c: Q)
'0 :;::
~0.05 (.)
P ....
II'
o a 6.
a = 20° a = 11° a = 8° Theory [7]
~'" -- ~"'a=20° ... Q .. -.... 10" -
0 ---- a = 11° p..- r"'--._.0- 1'"----- --.-~ ... ---- a= 8°
0.1 0.2 0.3 0.4 Cavitation number (J'
Fig. 19 - Lift Coefficient as a Function of Cavitation Number, C. = 0, AA = 2.5
I
~O.IO (.) -E eO.05 )
0 0
0 a 6.
0.1 0.2 0.3
0.6
0.5
.J (.)
-0.4 -:!: b 'EO . Q)
'0 :;:: -Q)
8 0 .
o
3)
2
.I
0 o
..... o ... ' 1-'11 =20° , ... a
0 •• 10' , ...
,1:t=uo ,,'
~ .... "
, .. ... ,'a=8° .. n ... ' .. ' ... ' " 0 .. " .. '
"," ~~ ,.
.' -' " .. :; ..
-'"
0.1 0.2 0.3 0.4 Cavitation number (J'
Fig. 20 - Drag Coefficient as a Function of Cavitation Number, C. = 0, AA = 2.5
I
(;)
a
0.4 0.5 0.6 0.7 Cavitation number (J'
Fig. 21 - Moment Coefficient as a Function of Cavitation Number, Cj = 0, AA= 2.5
55
56
• ...1 0.151--+---i---:--b,...w.n+--If--
~ i 0.1 OI---;-.,7'---I-r--'IS+----+-
.!!! u ::: 0 .05t---.,~-+.--I!!IH----LII) o o _ O.OC!----'--~ -
0.00
rY <l~0.025
..: c: II)
'0 ;;: 0.000 .... II) o o~0.025
CII c ... o .!:
+
+
0 0 0 0
IJ. 8 a =8"
12J'O + IJ. £IIILlQC
0 0
IJ.
oW L<JlJ
0
~ IJ. C 0
-c:
+':7""-'--t-......... --;1J. C 0.025 + +
IJ. q=IIO
~ 0.0,5I--+-!---+---I
~ ~~--~~--+---~
u .E 0.01(11---'--
a=2
0.015l-+-\----'=+---\---+---\---+----I
0.02 004 006 0.08 010 012 014
Jet'Momentum Coefficient; CJ
Fig. 22 - Increment in Lift Coefficient, AA = 2.5
Jet Flap Angle, T =33°
o 0' - 0 o 0' Intermed iote IJ. 0'> 0.4, a =200
IJ. 0'>03, a=llo
IJ. 0' > 0.'1, a= 8°
II)
e II) ... u .E
0
~002
~o.
a=2
0 0
0
O' 002 004 0.06 0.08 010 0.12 0.14
Jet Momentum Coefficient, CJ
Fig. 23 - Increment in Drag Coefficient, AR = 2.5
~005
~0.04
::E o <l ~0.03 -c: II)
'0 ~0.02 ;;:: .... II)
<3 ~0.01 -<:
~ ~0.03 o ~
c: -002 -<:
E -0.01 II) ... u
.E -0.03
-002
IJ., A
a=aJ IJ.
A
o [ nO 0
0
0 lOJ IJ.
IJ. Olf!! Ol~
00 t. a=I'lo
0 ( 0 -
0 0 0 IJ.
0 0
IJ.
P 0
d 'uq a=20
0 t'"' Dc;:! L ~
P 0 0
-001 0 0.02 004 0.06 ooa 0.10 0.12 0.14
Jet Momentum Coefficient, CJ
Fig. 24 - Increment in Moment Coefficient, AR = 2.5
1.0r---r~--r-::o::---;"'----A--"'---;--T"o.--T=-_ -Z-O.-/SI.=--co-' 0.8 r.-t--l°r;;l,-HJ..-j~4--=~-+----I 0--0 0<0-<0; a>o; 00-
0 0 /:;
1.2
~!I chord a>O O«r<Cf a'>q
in.
/:;-{ 36 2.00 0 0 /:; .rr ru ;o"n-< R, 38 2.75 -0- ..[]- -tr
~ n ~ 'a= 8° I I I OJ=02
1,0
.... 0.8
Coefficient, t:.J
0.,26-'-"0.1;;0;;'2 'WmO"4i"'"-nl< UJIj;;-n';;, (;08"'Q.t.lo.,.......?to.t.;12i"'"i"'"Q,.,.Ior4---;Q<'oI6;--'7'I;;;~ Jet Momentum CoeffiCient, CJ
12 d, .=33· Al=2!:
t;) .A 0'-0. o.<uoOOj O>Oj '-'
a.U~ 0 0 /:; 0 hQJ 0 /:; I I
a=8" I I OJ=O.1
1.0
0,8
i 06
i J,6----'-1 ~l J 0----1---,.' t~ .1...--..:1..-1 0 ~:[3---,----,-1 1---,----,1 1
57
i:i llotlj~)rl Jl1 Jet Momentum Coefficient, CJ
Fig. 25 - Thrust Recovery
I
APPENDIX A --------NOTATION
!fE~li~l! A Notation
JllTany of the following terms are defined in Fig. A-l as well as in the text where they first occur.
fR - aspect ratio
b - width of jet flap at downstream infinity B - width of free jet at upstream infinity c - chord length of flap plate hydrofoil
CD - drag coefficient, Eq. (6)
C. - jet momentum coefficient, Eq. (2) J
CL - lift coefficient, Eq. (5)
CM - moment coefficient, Eq. (7)
CT - jet flap thrust coefficient CT = T/(cspu2/2)
D - drag including thrust from jet flap, if any (measured in direction of flow at upstream infinity) , D - force measured by dynamometer in plane through beams
e - width of cavity re-entrant jet measured at speed U k - proportionality constant ~ - cavity length (measured from leading edge of hydrofoil) L - lift (measured normal to flow at upstream infinity) ,
L - force measured by dynamometer normal to plane tbroug~ beams m - distance from geometrical center of dynamometer to mid chord of foil M - moment (measured about mid chord, positive nose up)
Pro - pressure surrounding free jet
Pc - measured cavity pressure
Pv - vapor pressure of tunnel water
Q - total volumetric discharge of jet flap QA - volumetric air supply rate to ventilated cavity
s - span of hydrofoil T - thrust from jet flap (measured in direction of flow at upstream in.finity)
U - tunnel speed ~ - distance from mid chord to center of pressure, positive toward leading edge
62
V. - jet flap speed at the slit J
Vro - jet flap speed at downstream infinity
a - angle of attack (measured between hydrofoil chord and vertical flow
axis of tunnel)
t
a - angle between plane of beams in dynamometer and vertical flow axis of
tunnel
~ - de~lection angle of free jet produced by lifting body
o - width of jet flap after emergence from slit
ACD - increment in drag coefficient attributable to jet flap
ACL - increment in lift coefficient attributable to jet flap
ACM - increment in pitching moment coefficient attributable to jet flap
8~ - increment in center of pressure attributable to jet flap
c
~ - jet flap angle (measured between hydrofoil chord and tangent to jet
flap at slit)
p - water density
cr - cavitation number, Eq. (3)
cr - reference cavitation number, Eq. (4) v
c-_~
Fig. A-l - Definition Sketch
Reentro.nt jet
Pco
~ b --Val
B+I
)
!EE§l!12f! B THE JET-FLAP DYNAMOMETER
!E.E.~li12.~! B The Jet-Flap Dynamometer
Several dynamometers have been constructed and installed in the twodimensional, free-jet water tunnel, but none of those previously constru.cted permitted insertion of a source for a jet flap_ Hence, still another dynamometer was required for this series of experiments. This new dynamometer is described in this appendix.
As in the past, two identical dynamometer units are involved, one for the front face and one for the rear face of the tunnel. Each unit is built around a heavy brass cup which fits into 3.7-in. diameter holes in opposite faces of the test section. The joint is sealed with an "0" ring. Figure 2 is a photograph of the unit installed on the front face, while Fig. 3 shows photographically some details of a dynamometer unit and test bodies.
Each unit is of the displacement type, small displacements being measured by strain gages cemented to tension fibers. The fibers, located just outside the tunnel, support main beams which are otherwise cantilevered from a rigid support frame farther outside the tunnel and which, at their opposite ends, support a test body inside the tunnel.
There are two main beams in each unit; these are pipes. Fluid for the jet flap flows through one of these, while the other may be used for measuring cavity pressure or for ventilating a cavity. (The pipes can also be used for housing levers to manipulate solid flaps or for other purposes.) The flexible hose carryi.ng water for the jet flap, visible in Fig. 2, is attached to the lower pipe beam at its fixed end. Consequently, there is no influence on the dynamometer reading because of variable flow rate or angle of attachment of the hose; this was verified by running water through the beam at various rates without a test body and npting that the dynamometer shp~ed no deflection.
The beams pierce the bases of the dynamometer cups through holes with large clearance. Each beam at its free end is fastened to a copper bellows as may be seen in Fig. 3bo The bellows are attached to the ba.ses of the dynamometer cups at their other ends and serve to seal the large holes. The bellows are fixed in length by the pipe beams but deflect rather readily in
66
the lateral direction as the beams bend. The differential pressure between
the inside and outside of the bellows has no influence on their lateral de
flective properties so that the dynamometer may be calibrated with dead weights
while the tunnel is dry.
For testing finite-span bodies only one dynamometer unit is used,
while the hole in the opposite test section wall is closed with a transparent
plug (see Fig.S). Two.dimensional test bodies are suspended between the two
dynamometer units and are of such length that there is end play of about 0.040
in. when they are instfl-lled in the dry tunnel. This permits the tunnel walls
to be.BUcked together without putting an axial strain on the pipe beams when
vapor pressure exists inside the tunnel. The bodies are rigidly mounted on
3025-in. diameter end disks (spaced S.06-in. apart in the case of two-dimen
sional bodies, S.06-in. being the distance between tunnel walls at the test
section with vapor pressure inside the tunnel). The end disks fit just in
side the dynamometer cups and close the cups with about O.OOS-in. radial
clearance under no-load conditionso The disks form smooth continuations of
the tunnel walls when the tunnel is operating at vapor pressu!e and extend
slightly into the tunnel in the two-dimensional case at higher pressures.
Typical hydrofoil test bodies mounted for use with this dynamometer
system are shown in Figs. 3c and 3d. The small cups on the reverse side of
each disk from the foil slip over the pipe beams of the dynamometer with an
"0" ring seal and are the only connection between dynamometer and test body
in the two-dimensional case. In the finite span case the single end disk has
to be held to the base of the dynamometer cup by two tensioned wires; these
pull the disk tightly against the pipe beams with constant axial load on the
beams but do not contribute to the lateral deflection.
Two-dimensional test bodies are installed by withdrawing one dyna
mometer unit far enough to clear the cups. Angle of attack is set by rotat
ing both dynamometer· units together in such a manner that there:I;s zero moment
under no-load conditions. They are turned by hand against "0" ring friction
and turn rather easily when the tunnel is not operating. Tunnel suction draws
the dynamometer units in tightly during operation and prevents turning as well
as enhances sealing. At zero angle of attack the pipe beams lie in a. nea.rly
vertical plane.
67
The fibers supporting the pipe beams in each unit are flat pieces * of steel shimstock, 3!8-in. wide with 3/4-in. free length; thickness depends
on the load to be carried. They occur in opposite pairs, only one member of each pair being visible in Fig. 3a, and are prestressed so that all fibers are continuously in tension regardless of loading. (This means that after pretensioning, the test body end disks no longer have a uniform radial clearance from the dynamometer cups, the clearance becomes uniform at some intermediate load, and is again non-uniform in the opposite sense at maximum load.) There are two SR-4 strain gages on each fiber, one on each face. These are combined electrically in a Wheatstone bridge with each other and with the pair of gages , from the opposite fiber to give a single nominal drag reading D and two
t " nominal lift readings, L U for the upper beam and L L for the lower beam for each dynamometer unit, six readings in all f'or the system. The bridges are balanced, and output voltages are read on Consolidated carrier amplifiers, one type 1-118 and one type 1-127 being available for the purpose. Typical t t t calibration curves for D L U' and L L' including interaction effects, , are shown in Fig. B-1. These were obtained using deadweights in the dry tun-nel. The forces shown on the graphs are total forces since it was at first assumed that both dynamometer units would behave identically and that readings would need to be taken from only one to get total effects.
True drag, lift, and moment measured by each unit were obtained by t . resolution knowing the angle a, between the nominal drag axis of the dyna-
mometer and the vertical flow axis. The formulas given as Eqs. (.5), (6), and (7) in the body of the report were used for this purpose. Forthe 2-in.-chord, , flat-plate foils typified by Fig. 4, a, is larger than the true angle of at-tack a. by 5.2 degrees; for the 2.75-in.-chord plate, it is larger by 3.9 degrees.
The dynamometer units in use are designed for 70 Ib on the upper lift beams, 40 Ib on the lower lift beams, and 20 Ib in drag (total loads). The capacity is easily changed by installing new fibers of different thickness. Overloadi.ng is automatically prevented because the test body end disks will rest on the dynamometer cups when the load becomes excessive.
* The fibers are clamped to a collar around the beams at one end and to the supporting structure at the other; a fine wire placed between the fiber and. clamp prevents slipping and determines the free length of the fiber.
68
Although force and moment data for computational purposes were read
directly from the amplifier dials as time averages, some data were sampled by
recording galvanometers. Figure B-2 shows a typical record from the rear face
dynamometer unit as recorded by a Sanborne Model 293 galvanometer. (The fluc
tuation scales shown on the figure actually vary with load because of the in
teractions.) . These records and those obtained at much higher paper speeds O.h
a Consolidated recorder (up to 85 times as fast) indicate that there are gen
erally no resonance problems with the dynamometer and test body up to fre
quencies of several hundred per second. (The flutter-like fluctuations near
the left edges of the diagrams in Fig. B-2 are produced by the clearing of
air from the jet flap pipe as the flap is started. There was one operating
condition that produced a lift vibration at 16 cycles per second with the jet
flap not operating, and this was avoided in taking data.)
Even though it was originally intended to take readings for two
dimensional hydrofoils from only one dynamometer unit, it was soon found that
moment was taken unequally between the two units (there being four points of
support, of which one is redundant). It thus became desirable to take data
from both units and this has been done throughout. The total lift and drag
given by each unit should be identical; this was not quite true. Maximum
discrepancies of the order of 10 per cent have occurred between the units and
these were considered to be acceptable. Discrepancies may be attributable
to several factors; these include the occasional transfer of moment support
from one side to theother,non-symmetrical placement of the test body in the
tunnel because of the end play permitted between dynamometer units, and, per
haps, a slightly non-symmetrical flow in the tunnel. The plotted data were
obtained by averaging the data obtained from the two units.
The most serious operating problem with this dynamometer has been
caused by temperature differences acting on the strain gages. The strain gages
are open to the atmosphere surrounding the tunnel and are normally at room
temperature whether or not the tunnel is operating. However, the jet flap is
operated from the citywater mains which maintain water at a temperature less
than room temperature. Since the fibers are metal and are attached directly
to the brass pipe beam carrying city water, the fiber and strain gage tem
peratures are altered when water is turned off and on. To overcome this dif
ficulty, the dynamometers were calibrated and the amplifiers were zeroed, at
69
first with the jet flap water running but discharging axially from the pipe
beams (no test body). Later, itwas found that zeroing could be done equally
well with a test body in place by first running the jet flap fluid at a high
rate to cool the pipe and then running it at a trickle during zeroing to keep
the pipe cool, the trickling fluid produced .no measurable force. Furthermore,
the rear dynamometer unit is activated with bakelite temperature-compensated
gages, and the other unit, which uses paper gages, is placed in the stream of
a continuously running fan to equalize temperatures. Since the operating
practices described in this paragraph have been pursued, no temperature dif
ficulties have been encountered.
70
700
600
500
400
300
200
/00
.......::
~ V ./
~L·20 Ibs.~ ~
£ J~ ~[L ·0 Ibs.
~ ~.
~ v
0 20 40 60 80 100
-~-~~ ~
L.U (Pounds)
-'-
350 Q) - 300
c. 250 E
« 200
c 0 /50 0\ c 100 "0 C 50 Q)
0:::
Q) 0
c 0
10 20 30 40 50
L.L {Pounds}
en
350
300
250
200
150
100
50
~ j:7
[u· 42 Ibs. /. ~ [L= 19 IbS~ V
./ ~ V,[u· 0 Ibs . r----
.........: t:/ [L· 0 Ibs .
~ V
~ v
o 4 8 12 16 20
D' (Pounds)
Fig •. B-1 - Typical Calibration Curve for Dynamometer
71
/Fluctuation Scales
+- .,-
Gv +'-
38.6 fbs.
1.5 fbs.
-8.0 Ibs,
0'
o Ibs.
Foil No.3
Q: II deg.
Ci: 0.194 Time ~ I sec. Cj= 0.086 (max.)
Fig. B-2 - Sample Record from Rear Dynamometer Unit
APPENDIX C .,.....~----- .... EFFECT OF FREE-JET DEFLECTION
Effect of Free-Jet Deflection
A momentum balance of the flow shown in Fig. A-l yields, per unit
width,
L+ 2 pU e sin ~ - pLfB si!ll. ~ - V2 b p co sin !3 = 0 (C-l)
D - pU2B 2 cos !3 + pU2B cqs ~ + V2 b cos 13 0 pU e p co = (C-2)
Here Ue is the flow rate of fluid in the re-entrant jet within the cavity
which must eventually be re-accelerated to the 'velocity U by the remainder
of the stream. (Or, more conventionally, Ue is the strength of a sink with
in the cavity.)
Solving Eq. (C-l) for sin ~,
L CL sin 13 = --~--~~-------- = --~--~--------
pU2(B - e) + p~ b 2(~ - ~) + -be CJ. ro c c
Assuming that e«B and that b ~ 6
CL ~ ~ sin ~ ~ ----~------
2(B/c) + C. J
(v ~ V.), ro J
(C-3A)
(C-3B)
since 2B/ c ~10 and CL < 1; with C j« 2B/ c, ~ ~ cCL/2B- 0.1 rad. or less.
Now Eq. (C-2) may be solved for D to yield
e B 6 (c-4) C = 2- cos 13 + 2 -C- (1 - cos 13) - b Cj cos 13 D c
~2 _e_ + B ~2/c ° c - bCj
form drag induced drag thrust
76
The thrust term may be written
(C-5
so that if b ~ 6, CT = Cj cos 13 ~ Cj for 13«1.
, .' '.~
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-~--~~,-~.--,~--.-"--.-- -
Technical Paper No. 46, Series B St. Anthony Falls Hydraulic Laboratory EXPERIMENTS ON A JET FLAP IN SUPER CAVITATING FLOW, by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710 (50).
Experiments on a fully-cavitated, flat-plate hydrofoil equipped with a pure jet flap were conducted in the free-jet water tunnel at the St. Anthony Falls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift, drag, and moment coefficients and on the shift in center of pressure as a function of jet momentum coefficient, cavitation number, and other variables. The experimental results were qualitatively similar to those that are obtained in fully wetted flow but the increments are similar quantitatively.
AVailable from St. Anthony FaIls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
Technical Piper No. 46, Series B St. Anthony FaIls Hydraulic Laboratory EXPERIMENTS ON A JET FLAP IN SUPER CAVITATING FLOW, by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710(50).
Experiments on a fully-cavitated, flat-plate hydrofoil equipped with a pure jet flap were conducted in the free-jet water tunnel at the St. Anthony FaIls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift, drag, and moment coefficients and on the shift in center of pressure as a function of .jet momentum coeffiCient, cavitation number, and other variables. The experimental results were qualitatively similar to those that are obtained in fully wetted flow but the increments are similar quantitatively. .
Available from St. Anthony FaIls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
1. Supercavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
I. Title n. Silberman, Edward
m. St. Anthony FaIls Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
1. Supercavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
I. Title n. Silberman, Edward
m. St. Anthony Falls Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
Technical Paper No. 46, Series B St. Anthony Falls HydrauliC Laboratory EXPERIMENTS ON A JET FLAP IN SUPER CAVITATING FLOW, by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710(50).
Experiments on a fully-caVitated, flat-plate hydrofoil equipped with a pure jet flap were conducted in the free-jet water tunnel at the St. Anthony Falls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift, drag, and moment coefficients and on the shift in center of pressure as a function of jet momentum coefficient, cavitation number, and other variables. The experimental results were qualitatively similar to those that are obtained in fully wetted flow but the increments are similar quantitatively.
Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
Technical Paper No; 46, Series B St. Anthony Falls Hydraulic Laboratory EXPERIMENTS ON A JET FLAP IN SUPER CAVITATING FLOW, by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710(50).
Experiments on a fully-caVitated, flat-plate hydrofoil equipped with a pure jet flap were condac.ted in the free-jet water tunnel at the St. Anthony Falls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift, drag, and moment coefficients and on the shift in center of pressure as a function of jet momentum coeffiCient, cavitation number, and other variables. The experimental results were qualitatively simHar to those that are obtained in fully wetted flow but the increments are simHar quantitatively.
Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
1. Supercavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
I. Title II. Silberman, Edward
III. St. Anthony Falls Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
1. Super cavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
1. Title n. Silberman, Edward
m. St. Anthony Falls Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
Technical Paper No. 46, Series B St. Anthony Falls Hydraulic Laboratory EXPERIMENTS ON A JET FLAP IN SUPERCAVITATING FLOW, by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710 (50).
Experiments on a fully-cavitated, flat-plate hydrofoil equipped with a pure jet flap were conducted in the free-jet water tunnel at the St. Anthony Falls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift, drag, and moment coefficients and on the shift in center of pressure as a function of jet momentum coefficient, cavitation number, and other variables. The experimental results were qualitatively similar to those that are obtained in fully wetted flow but the increments are similar quantitatively.
Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
Technical Paper No. 46, Series B St. Anthony Falls Hydraulic Laboratory EXPERIMENTS ON A JET FLAP IN SUPERCAVITATING FLOW, by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710 (50).
Experiments on a fully-cavitated, flat-plate hydrofoil equipped with a pure jet flap were conducted in the free-jet water tunnel at the st. Anthony Falls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift. drag. and moment coefficients and on the shift in center of pressure as a function of .jet momentum coefficient. cavitation number. and other variables. The experimental results were qualitatively similar to those that are obtained in fully wetted flow but the increments are similar quantitatively. .
Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
1. SUpercavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
I. Title II. Silberman, Edward
III. St. Anthony Falls Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
1. SUper cavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
I. Title II. Silberman, Edward
III. St. Anthony Fails Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
Technical Paper No. 46, Series B St. Anthony Falls Hydraulic Laboratory EXPERIMENTS ON A JET FLAP IN SUPERCAVITATING FLOW. by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710(50).
Experiments on a fully-cavitated, flat-plate hydrofoil equipped with a pure jet flap were conducted in the free-jet water tunnel at the St. Anthony Falls Hydraulic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift, drag, and moment coefficients and on the shift in center of pressure as a function of jet momentum coefficient. cavitation number. and other variables. The experimental results were qualitatively similar to those that are obtained in fully wetted flow but the increments are similar quantitatively.
Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
Technical Paper No; 46. Series B St. Anthony Falls Hydraulic Laboratory EXPERIMENTS ON A JET FLAP IN SUPER CAVITATING FLOW, by Edward Silberman. January 1964. 78 pages incl. 28 illus. Contract Nonr 710 (50).
Experiments on a fully-cavitated. flat-plate hydrofoil equipped with a pure jet flap were conducted in the free-jet water tunnel at the St. Anthony Falls HydraUlic Laboratory. The foils were of 2-in. and 2.75-in. chord and were tested in 6-in., 10-in., and 14-in. wide free jets. Data were obtained on the increments in lift, drag, and moment coefficients and on the shift in center of pressure as a function of jet momentum coefficient, cavitation number, and other variables. The experimental results were qualitatively simHar to those that are obtained in fully wetted flow but the increments are simHar quantitatively.
Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $2.50 per copy.
1. SUper cavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
I. Title II. Silberman, Edward
IIi. st. Anthony Fails Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
1. SUper cavitating Flow 2. Jet Flap 3. Hydrofoil 4. Steady Flow
I. Title II. Silberman, Edward
III. St. Anthony Falls Hydraulic Laboratory
IV. Contract No. 710(50)
Unclassified
