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Page 1: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 2: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 3: on a Jet Flap in Supercavitating Flow - Conservancy

Reproduction in whole or in part is permitted

for any purpose of the United States Government

Page 4: on a Jet Flap in Supercavitating Flow - Conservancy

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

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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

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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

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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

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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.

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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

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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

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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

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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

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, 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

Page 14: on a Jet Flap in Supercavitating Flow - Conservancy

~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)

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(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)

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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

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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.

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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.

Page 19: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 20: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 21: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 22: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 23: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 24: on a Jet Flap in Supercavitating Flow - Conservancy

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)

Page 25: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 26: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 27: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 28: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 29: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 30: on a Jet Flap in Supercavitating Flow - Conservancy

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-

Page 31: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 32: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 33: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 34: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 35: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 36: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 37: on a Jet Flap in Supercavitating Flow - Conservancy

IlQQR~§'

(1 through 25)

Page 38: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 39: on a Jet Flap in Supercavitating Flow - Conservancy

33

Fig 0 2 - Front Face of Tunnel

Page 40: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 41: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 42: on a Jet Flap in Supercavitating Flow - Conservancy

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-

Page 43: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 44: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 45: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 46: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 47: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 48: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 49: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 50: on a Jet Flap in Supercavitating Flow - Conservancy

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 +

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

Page 51: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 52: on a Jet Flap in Supercavitating Flow - Conservancy

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:

Page 53: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 54: on a Jet Flap in Supercavitating Flow - Conservancy

~~~~~-~~~~~ =:;:-=-~~~::-=~~~~~-"--"-----==:::-=-=-=--==~---=- ==-------=--'-~-=--- - ------

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

Page 55: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 56: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 57: on a Jet Flap in Supercavitating Flow - Conservancy

-.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

Page 58: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 59: on a Jet Flap in Supercavitating Flow - Conservancy

-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,·:

Page 60: on a Jet Flap in Supercavitating Flow - Conservancy

=-::::'-:~---::-';:::::-;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:..

Page 61: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 62: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 63: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 64: on a Jet Flap in Supercavitating Flow - Conservancy

APPENDIX A --------NOTATION

Page 65: on a Jet Flap in Supercavitating Flow - Conservancy

!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

Page 66: on a Jet Flap in Supercavitating Flow - Conservancy

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

)

Page 67: on a Jet Flap in Supercavitating Flow - Conservancy

!EE§l!12f! B THE JET-FLAP DYNAMOMETER

Page 68: on a Jet Flap in Supercavitating Flow - Conservancy

!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

Page 69: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 70: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 71: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 72: on a Jet Flap in Supercavitating Flow - Conservancy

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.

Page 73: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 74: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 75: on a Jet Flap in Supercavitating Flow - Conservancy

APPENDIX C .,.....~----- .... EFFECT OF FREE-JET DEFLECTION

Page 76: on a Jet Flap in Supercavitating Flow - Conservancy

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

Page 77: on a Jet Flap in Supercavitating Flow - Conservancy

76

The thrust term may be written

(C-5

so that if b ~ 6, CT = Cj cos 13 ~ Cj for 13«1.

Page 78: on a Jet Flap in Supercavitating Flow - Conservancy

, .' '.~

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Commanding Officer and Director, U. S. Naval Civil Engineering Lab­oratory, Port Hueneme, California, Attn: Code L54.

j )~_.Bavidson Laboratory, Stevens Institute of Technology, Hoboken, New ,

I

v{ Jersey.

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·1 Dr. A. V. Hershey, Computation and Exterior Ballistics Laboratory, U. S. Naval Weapons Laboratory, Dahlgreen, Virginia.

\/1 Superintendent, U. S. Naval Academy, Annapolis, Maryland, Attn: Li­brary.

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80

Copies Organization

2 U. S. Mari time Administration, GAO Building, 441 G Street, N. W.,

Washington, D. C. Attn:

1 - Division of Ship Design

1 - Division of Research

v i 3 California Institute of Technology, Pasadena 4, California, Attn:

Jr1.:' ~ Professor M. S. Plesset

1 - Professor T. Y. WU

1 - Professor A. J. Acosta

1/1 University of California, Department of Engineering, Los Angeles 24,

California, Attn: Dr. A. Powell.

: /2

/1 I

Harvard University, Cambridge 38, Massachusetts, Attn:

1 - Professor G. Birkhoff (Dept. of Mathematics)

1 - Professor G. F. Carrier (Dept. of Mathematics)

j/University of Michigan, Ann Arbor, Michigan, Attn: Professor R. B.

Couch (Dept. of Naval Architecture).

ji2 New York University, Institute of Mathematical Sciences, 25 Waverly

Place, New York 3, New York, Attn:

1 - Professor J. Keller

t~~ 1 - Professor J. J. Stoker

j3 The John Hopkins University, Department of Mechanical Engineering,

Baltimore 18, Maryland, Attn:

/// 1 - Professor S. Corrsin

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\/1 ~> Massachusetts Institute of Technology, Department of Naval Archi­

tecture and Marine Engineering, Cambridge 39, Massachusetts, Attn:

Professor M. A. Abkowitz, Head.

,~/ 2 Dr. G. F. Wislicenus, Ordnance Research Laboratory, Pennsylvania

State University, University Park, Pennsylvania, Attn: Dr.M. Sevik

·1 Netherlands Ship Model Basin, Wagenigen, The Netherlands, Attn: Dr.

Ir. J. D. van Manen.

1 Boeing Airplane Company, Seattle Division, Seattle, Washington, Attn:

Mr. M. J. Turner.

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2. Convair, A Division of General Dynamics, San Diego, California, Attn: 1 - Mr. R. H. Oversmith 1 - Mr. H. T. Brooke

/1 Rand Developement Corporation, 13600 Deise Avenue, Cleveland 10, Ohio, Attn: Dr. A. S. Iberall.

:,1 I Dr. Blaine R. Parkin, Air Research Manufacturing Corporation, 98.51-9951 Sepulveda Boulevard, Los Angeles 45, California.

The Rand Corporation, 1700 Main Street, Santa Monica, California, Attn: Technical Library.

'2 Stanford University, Department of Civil Engineering, Stanford, California, Attn:

a

1 - Dr. Byrne Perry 1 - Dr. E. Y. Hsu

Dr. Hirsch Cohen, IBM Research Center, Post Office Box 218, Yorktown Heights, New York.

Commanding Officer, Office of Naval Research Branch Office, 207West 24th Street, New York, New York 10011.

Oceanics, Incorporated, Technical Industrial Park, Plainview, L. I., New York, Attn: Dr. Paul Kaplan.

Engineering Experiment Station, Kansas State University, Manhattan, Kansas 66.504, Attn: Prof. D. A. Nesmith.

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Therm Advanced Research, Inc., 100 Hudson Circle, Ithaca, New York, Attn: Mr. Gary R. Hough.

Page 83: on a Jet Flap in Supercavitating Flow - Conservancy

-~--~~,-~.--,~--.-"--.-- -

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

Page 84: on a Jet Flap in Supercavitating Flow - Conservancy

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


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