Theoretical and experimental investigation of arbitrary aspect ratio, supercavitating hydrofoils operating near the free...

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Title:
Theoretical and experimental investigation of arbitrary aspect ratio, supercavitating hydrofoils operating near the free water surface
Series Title:
NACA RM
Physical Description:
94 p. : ill. ; 28 cm.
Language:
English
Creator:
Johnson, Virgil E
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Hydrofoils   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: The existing two-dimensional theory for the characteristics of supercavitating hydrofoils operating at zero cavitation number is modified to include the effects of camber, aspect ratio, and depth of submersion. For a comparison with the theory, two aspect-ratio-1 hydrofoils, a flat plate and one with a cambered lower surface, were tested by operating them near the water surface so that their upper surfaces were completely ventilated. The data obtained verified the theory to within 3 percent. Some experimental data are also presented for the models operating at nonzero cavitation number for both air and water-vapor-filled cavities.
Bibliography:
Includes bibliographic references (p. 51-52).
Additional Physical Form:
Also available in electronic format.
Statement of Responsibility:
by Virgil E. Johnson, Jr.
General Note:
"Report date December 12, 1957."
General Note:
"Declassified May 16, 1958"

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003834456
oclc - 150534081
System ID:
AA00009189:00001


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NACA RM L57116

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

RESEARCH MEMORANDUM


THEORETICAL AND EXPERIMENTAL INVESTIGATION OF

ARBITRARY ASPECT RATIO, SUPERCAVITATING

HYDROFOILS OPERATING HEAR THE

FREE WATER SURFACE

By Virgil E. Johnson, Jr.


SUMMARY


Theoretical expressions for the lift, drag, and center of pressure
of supercavitating hydrofoils of arbitrary section, aspect ratio, and
depth of submersion are developed for the zero-cavitation-number case.

In order to verify and supplement the theoretical investigation,
an experimental investigation was made in Langley tank no. 2 of two
aspect-ratio-1 hydrofoils, one with a flat and one with a cambered lower
surface. Zero cavitation number was obtained in the tank by operating
the hydrofoils near the water surface so that their upper surfaces were
completely ventilated. Data were also obtained at cavitation numbers
greater than zero for flows with vapor-filled cavities and with air arti-
ficially introduced to simulate the same flows at lower speeds. The tests
covered a range of water speeds up to 80 feet per second, angles of
attack up to 200, and depths of submersion from 0- to 85-percent chord.

The theoretical force and moment coefficients agreed with the data
obtained in the zero-cavitation-number tests within an accuracy of about
5 percent. The ventilated force and moment coefficients of both models
were independent of speed. As predicted by the linear theory of Tulin
and Burkart, the cambered hydrofoil lift-drag ratio was superior to that
for the flat plate at the same lift coefficient. Similar to the influ-
ence of camber on a fully wetted airfoil, the influence of camber in
ventilated flow was to effectively increase the angle of attack without
appreciably changing the lift-curve slope. Comparison of the present
results with the results of other investigations revealed that a sharp
leading edge is necessary for good efficiency in supercavitating flow.






NACA RM L5716


The nonzero-cavitation-number tests confirmed the finding of others
that the lift of a hydrofoil operating in a finite cavity depends only
on the cavitation number and is independent of whether the gas in the
cavity is air or vapor.


INTRODUCTION


The desirability of using auxiliary lifting surfaces such as hydro-
foils or hydroskis for reducing seaplane-hull loads and improving rough-
water performance is well established. Although the higher aspect-ratio
submerged hydrofoils with their accompanying high efficiencies could be
the most desirable lifting surfaces to be used, only the low-aspect-ratio
planing hydroski has so far been successfully applied as landing gear to
modern high-speed aircraft. The reason for this is that the conventional
hydrofoil presents problems not experienced by a hydroski.

As a hydrofoil nears the free water surface (during a take-off run)
the low pressure side of the hydrofoil becomes ventilated from the atmos-
phere. This phenomenon results in a severe and usually abrupt loss in
lift and reduction in the lift-drag ratio. For conventional airfoil
sections, the loss in lift may exceed 75 percent. The speed at the
inception of ventilation depends on the angle of attack and depth of
submersion; but, except for very small angles of attack and relatively
low take-off speeds, the speed at inception is usually well below the
take-off speed of the aircraft.

Even if the ventilation problem is overcome by using small angles
of attack and incorporating fences or other devices for suppressing
ventilation, the onset of cavitation presents a second deterrent to the
use of conventional hydrofoils at high speeds. The loss in lift accom-
panying cavitation of conventional airfoil sections is not abrupt, but
the ultimate reduction in lift and lift-drag ratio is comparable to that
of ventilated flow. Even thin-airfoil sections of small design-lift
coefficient enter this cavitating regime of poor lift-drag ratios at
speeds in excess of about 80 knots.

Since the take-off speed of modern supersonic aircraft is in the
range of 150 to 200 knots, lifting surfaces with cavitating or venti-
lating characteristics superior to those of conventional airfoil sec-
tions are desirable. Fortunately, theoretical work by Tulin and Burkart
(ref. 1) has shown that superior configurations do exist and they have
selected a cambered configuration for operation in cavitating or venti-
lated flow which, in an inviscid fluid, has two-dimensional lift-drag
ratios at its design angle of attack and zero cavitation number, about
six times that of a flat plate. If such a cambered hydrofoil can be





NACA RM L5716 5


induced to ventilate at very low speeds, while the aircraft hull still
supports most of the load, a stable and efficient take-off run may be
possible.

The present report is concerned with an experimental and theoretical
investigation of one of the Tulin-Burkart cambered configurations and a
comparison of its characteristics with those of a flat plate. The pur-
pose of the investigation was (1) to find out if the superiority of the
cambered configurations obtained from two-dimensional, infinite-depth
theory would be maintained at finite depth and aspect ratio, and (2) to
develop expressions for the prediction of the force and moment character-
istics of a ventilated hydrofoil of arbitrary section profile which
would include these three-dimensional effects.


SYMBOLS


A aspect ratio

An,AO coefficients of sine-series expansion of vorticity distri-
bution on equivalent airfoil section at infinite depth,
that is,

W(x) = 2V (AO cot + A1 sin 9 + A2 sin 29 An sin no

where

2 P't d_
An = dy cos n9 dO



AO= 1 f dO +a =a.+Ao


An,h,AO,h coefficients of sine-series expansion of vorticity distribu-
tion on hydrofoil section

a distance from equivalent airfoil leading edge to center of
pressure in chords

Bn.BO coefficients of cosine series defining location of image
vortex in airfoil plane

b parameter defining location of spray at infinity in t plane
(see ref. 2)






NACA RM L57I16


CD total-drag coefficient, -2-
qS

CDi profile-drag coefficient (total drag less crossflow and
9 ^ D!
skin friction), D
qS

CD]c crossflow-drag coefficient (normal to the section), -C

Df
Cf skin-friction drag coefficient, --
qS

CL total-lift coefficient, --
qS

CL total-lift coefficient of equivalent airfoil section, -


qS
LL

CL 1 lift coefficient exclusive of crossflow, -

CLc e crossflow lift coefficient, C
qS

CM pitching-moment coefficient (about the leading edge), -
qSc

Cm pitching-moment coefficient of equivalent airfoil section
(about the leading edge), -
QSc

C_ 3 third-moment coefficient of equivalent airfoil section
(see ref. 1), -3


N qS
CM resultant-force coefficient on arbitrary section, S=


CN,f resultant-force coefficient of flat plate,
qS

CnCO coefficients of sine-series expansion of vorticity distri-
bution on equivalent airfoil section at arbitrary depth

c chord

D total drag force

D1 drag force excluding crossflow drag and skin friction






NACA RM L57116


DC drag force due to crossflow (normal to section)

Df drag force due to skin friction

d leading-edge depth of submersion

E Jones' edge correction, ratio of semiperimeter to span
(see ref. 14)

F resultant force

g acceleration due to gravity

L total lift force

LC lift force due to crossflow

L1 lift force exclusive of crossflow, L Le

2 distance from hydrofoil leading edge to stagnation point
in chords

M moment about leading edge
5
M5 third moment about leading edge, 2 p(i)x di

m = CL/a

p pressure, Ib/sq ft

PC pressure within cavity, lb/sq ft

pO pressure at mean depth of hydrofoil, Ib/sq ft

p fluid vapor pressure, lb/sq ft

q free-stream dynamic pressure, pV2

r distance from image vortex to point on equivalent airfoil
in chords

S area, sq ft

a span, ft

u perturbation velocity in X-direction






NACA RM L57116


V speed of advance, fps

v perturbation velocity in Y-direction

X,Y coordinate axes

x distance from leading edge along X-axis

x distance from leading edge to center of pressure

yl distance from the X-axis to hydrofoil lower surface

yu distance from the X-axis to hydrofoil upper surface

a geometric angle of attack, radians unless otherwise specified

ac angle-of-attack increase due to camber, (CO' + Ci C2/2)

or at infinite depth ac,= = (A, A2/2), radians unless
otherwise specified
m
ai induced angle of attack, radians unless otherwise specified

ao angle between hydrofoil chord line and reference line, posi-
tive when chord line is below reference line, radians
unless otherwise specified

a' angle of attack measured from hydrofoil chord line,
a' = a + mo, radians unless otherwise specified

P circulation, strength of single vortex

7 central angle subtending chord of circular-arc hydrofoil

b spray thickness at infinite distance downstream

e deviation of resultant-force vector from normal to hydrofoil
reference line

9 parameter defining distance along airfoil chord,
x = C(1 cos 9)
2
l b-sec2
p mass density, -ft4
ft4

a cavitation number, Po c
q


cavitation number at inception






NACA RM L57l16


T correction factor for variation from elliptical plan form

g angle between spray and horizontal

angle between i-axis and line joining image vortex with a
point on equivalent airfoil

L vorticity

I I indicates "function of" for example, CNHa = CN,fIa + ac.

Z = X + iY

S= tan-1 = M + C
CL

Subscripts:

e effective

0 zero depth of submersion

t total

W infinite depth of submersion

c.p. center of pressure

Barred symbols refer to equivalent airfoil section and unbarred
symbols refer to the supercavitating hydrofoil section.


DESCRIPTION OF SUPERCAVITATING FLOW


The parameter defining cavity flow is a = P where po is
the pressure at the mean depth, pc the pressure within the cavity, and
q., the dynamic pressure. The magnitude of a for the condition at which
cavitation is incipient is defined by the particular value oi. If a
is reduced below oi, cavitation becomes more severe; that is, the cavita-
tion zone extends over a larger area. When a hydrofoil operates at suf-
ficiently low values of a, the cavity formed may completely enclose the
upper or suction surface and extend several chords downstream as shown in
figure 1(a). Theoretically if the cavitation parameter is reduced to zero,
the cavity formed will extend to infinity. The flow regime where the
cavity length exceeds the chord is defined as supercavitating flow.






8 NACA RM L57I16


Supercavitating flow, may be obtained by either increasing velocity
or cavity pressure or both. At a constant depth and water temperature,
a and therefore, the length of the cavity is dependent only on the
velocity since po pc is then po Pv and is constant. If part or
all of the boundary layer of a configuration is separated, the eddying
fluid in the separated region can be replaced by a continuous flow of
lighter fluid such as air. (See refs. 5 and 4.) Regulation of the amount
of air supplied will control the cavity pressure and thus the length of
the cavity formed. If the quantity of air supplied is very large, the
cavity pressure will approach the ambient pressure po and a very long
cavity will result even at low-stream velocities.

The ventilation of surface piercing hydrofoils is therefore a super-
cavitating flow due to large quantities of air supplied from the atmos-
phere to separated flow on the suction surface of the hydrofoil. Super-
cavitating flow as a result of ventilation also occurs when a nonsurface-
piercing hydrofoil of moderate aspect ratio operates near the free surface
(see fig. 1(b)). As pointed out in reference 5, air is entrained in the
trailing vortices and drawn to the suction side of the hydrofoil causing
a long trailing cavity to enclose completely the upper surface of the
hydrofoil and extend far downstream. The ventilated type cavity described
in reference 5 differs in shape from those formed in deeply submerged flow
because of the proximity of the free water surface. It is similar to
planing, with the spray forming the upper surface of the cavity. Since
the cavity pressure is approximately the same as the ambient pressure
(at small depths of submersion), the cavitation number for this type
flow is nearly zero.

The experimental portion of the present investigation utilized both
of the methods of obtaining low values of a discussed in the foregoing
paragraphs; that is, varying velocity and cavity pressure. However,
the case of zero cavitation number was of particular interest since it
represents the condition of minimum lift and also because a major portion
of the existing cavity flow theory assumes this condition. Therefore,
most of this report treats the ventilated type supercavitating flow.


ZERO-CAVITATION-NUMBER THEORY


TWO-DIMENSIONAL THEORY


Flat Plate

The characteristics of a two-dimensional inclined flat plate in an
infinite fluid, operating at zero cavitation number, have been obtained





NACA RM L57I16


by Kirchoff and Rayleigh (ref. 6). The resultant force on the plate is
given by the well-known equation

CN,f = 21 sin a (1)
4 + e sin a

The distance from the leading edge to the center of pressure is


(x) .1 (2 3 cos a (2)
(x} 2 0 L\(2)
\
Similar work was performed by Green refss. 2 and 7) to include the
effect of the free water surface (but neglecting gravity). The solution
is necessarily obtained in terms of the spray thickness 5 rather than
the more useful depth of submersion and is given as two parametric equa-
tions in terms of the parameter b.

2(b b2 1)sin a cos a
CL = CNYf cos a = = mM (3a)


S_ b -cos a (5b)
c D
where

D = (b b2 1)sin a + 1T2 cos a + (b cos a l)loge

This result is plotted as the variation of m with 5/c for various
angles of attack in figure 2.

Since gravity is neglected in Green's solution, it is important to
understand the effect of the absence of this force on the result obtained.
If the Froude number based on chord V2/gc is large, then near the plate,
where the streamline radii of curvature are small, the inertia forces are
indeed large compared to the force of gravity. Thus, under these condi-
tions, equations (5) for the force on the plate should be applicable, for
a given 5/c, in spite of the neglect of gravity in its derivation. How-
ever, the variable 5/c is not usually known. It is desired to know the
force on the plate for a given depth of submersion; therefore, the rela-
tionship between the leading-edge depth of submersion and the spray
thickness is needed. The effect of gravity becomes very important in
determining this relationship. Because of the neglect of gravity in the
analysis of Green, the plate is always located infinitely above the still
water surface a physical impossibility in a real fluid. Gravity cannot
be neglected in determining the location of the free water surface. The
reason for this is that several chords away from the plate the streamline






NACA RM L57116


radii of curvature become very large and thus the inertia forces become
of the same or lower order of magnitude than the force of gravity. When
gravity is present the distance of the plate above the water surface at
infinity is at least limited to the dynamic pressure head. The effect
of finite aspect ratio is also to place the plate finitely near the free
water surface as will be discussed in a subsequent section.

In summary, if the Froude number is large, the forces on an inclined
flat plate near a free water surface can be obtained in terms of the spray
thickness 8/c from equations (5). The relationship between the spray
thickness and the actual leading-edge depth of submersion cannot be deter-
mined from Green's analysis, but the relationship is known to be influ-
enced by the presence of gravity and the aspect ratio of the plate. The
method by which the lines of constant depth (shown in fig. 2) were obtained
is discussed in a subsequent section.

The angle V that the spray makes with the direction of motion is
also obtained from reference 2 as


1 = cos-1( cos a -a (4)
T b cos a /



The variation of $ (eq. (4)) with 5/c (eq. (5b)) is shown in figure 5.


Cambered Sections

Cambered surfaces can theoretically be analyzed in two dimensions by
the method of Levi-Civita (ref. 7). However, like many conformal mapping
problems the method is very difficult to apply to a particular configura-
tion and only a few specific solutions have been obtained. Among these
is the work of Rosenhead (ref. 8) and Wu (ref. 9). Although the solution
of Wu is applicable in principle to arbitrary sections, the solution was
presented only for the circular arc. A particular advantage of Wu's
solution is that it includes the effects of nonzero cavitation number.

The most useful treatment of cambered surfaces is the linearized
theory of Tulin and Burkart (ref. 1) which is readily applicable to any
surface configuration (with positive lower surface pressures) as long
as the angle of attack and camber are small. The principal results of
this linearized theory are summarized below.

The supercavitating hydrofoil problem in the Z-plane is transformed
into an airfoil problem in the Z-plane by the relationship Z = -JZ. By





NACA RM L57116


denoting properties of the equivalent airfoil with barred symbols and
those of the hydrofoil with unbarred symbols, the following relationships
are derived:

=5) = y(2) (5)
di dx


U() = u(2) (6)



C=I= + A1 2 (7)
2

CD =t L) Ao k (8)
8t2' 2 )



Cm = Cm,53 = (5AO + 7A1 7A2 + 3A5 (9)


The coefficients An are the thin-airfoil coefficients in the sine-
series expansion of the airfoil-vorticity distribution


Q(R) = 2V(Ao cot + An sin no) (10)
\ ~n=1
where


S= (l cos 9) (0O 9 $ )
2

and can be found for a given configuration from the following equations


AO = 0 dO + a = a + AO' (lla)






NACA BM L57116


An = cos nG dG (11b)



By using equations (7) and (8), the condition that AO in equation (10)
is zero, that SI(R) does not contain harmonics greater than two, and
that 2(R) is everywhere positive over the chord (that is, positive
lower surface pressures on the hydrofoil), Tulin and Burkart (ref. 1)
obtained a low-drag family of hydrofoils. This particular family of
hydrofoils is given by the equation


y = A 312 ()2] (12)
c 2 c 3c) c


The lift-drag ratio of this foil at its design incidence (that is, a = 0)
is from equations (7) and (8)

L = ?5r (15)
D 4 R2CL


This 6.25-fold improvement over the L/D of a flat plate is most encour-
aging and was the impetus for this investigation.

The restrictions imposed by the assumptions of the Tulin-Burkart
theory prevents its use in the calculation of the characteristics of
hydrofoils suitable for use as aircraft landing gear. Here, because of
the high hydrofoil loads on necessarily thin hydrofoils the aspect ratio
may be as low as 1 or 2. Also the hydrofoil must operate near the free
water surface and in some instances at large angles of attack. Thus,
the effects of these variables on the characteristics of supercavitatfing
hydrofoils (particularly of cambered sections) is needed. Much of this
information can be obtained by certain modifications to the existing two-
dimensional theory discussed in preceding paragraphs.


MODIFICATIONS OF TWO-DIMENSIONAL THEORY


Nonlinear Equation for Lift at Infinite Depth

For any configuration the reference line of the section from which
the angle of attack is measured can be chosen such that AO' = 0. It is





NACA RM L5716


convenient in this report to assume that a = 0 refers to the reference
line which makes A0' = 0. With this assumption, equation (7) may be
written as


CL = (a + Al ) = j(a + ac) (14)


where ac is the effective increase in angle of attack due to camber

(Al -). Thus, the solution for cambered hydrofoils is merely the flat
plate linearized solution ra with a replaced by a + ac. This is
exactly analogous to the influence of camber on airfoils in an infinite
fluid where there is an effective increase in angle of attack due to the
camber. By carrying this procedure further, and by applying it to the
resultant force rather than the lift, the nonlinear solution of Rayleigh
becomes applicable to arbitrary configurations simply by replacing a
by a + ac; that is,

21 sin(a + c) (15)
4 + n sin(a + ac)



The lift will then be

2n sin(a + ac)
CL 3s---- cos p (16)
4 + A sin(a + ac)


Here, p = a + e, where e denotes the deviation of the resultant-force
vector from the normal to the hydrofoil reference line. For large values
of a, c is small compared with a and cos p T cos a. When a is
very small, about 00, e is a maximum and can be shown by use of equa-
tions (7) and (8) to be A- 2 if A0' = 0 as assumed. For any
4k(A, A)
practical low-drag hydrofoil, this value will almost always be less than
30 for which the cosine is very nearly 1 or cosine (a + c) cos a 1.
Therefore, cos 0 in equation (16) may be replaced by cos a with little
loss in accuracy and great gain in simplicity. Equation (16) then becomes

2n sin(a + ac)
CL 2 n- sin(a + cos a (17)
4 + it sin(a + ac)






NACA RM L57l6


For a circular-arc hydrofoil of central angle 7, it is shown in
appendix A that oc = (9/16)7 and that the reference line must be chosen
at an angle 7/8 to the chord line so that A0' = 0. The result obtained
by substituting this value into equation (17) is compared in figure 4
with the linear solution of Tulin and Burkart (eq. (7)) and the nonlinear
solution of Wu (ref. 9) for two circular-arc profiles. The agreement of
equation (17) with the more exact solution of Wu is good over the entire
range of angle of attack from 00 to 900. Similar agreement is expected
for any configuration of small camber.


Nonlinear Equation For Lift at Finite Depth

The successful modification of the Rayleigh equation to include
cambered configurations leads at once to a similar modification of the
solution of Green. However, in this case the argument for replacing a
by (a + ac) is very weak unless the section coefficients which deter-
mine o" are known as a function of the depth of submersion.

An examination of the linearized expressions for the lift coefficient
of arbitrary foils at infinite depth and at zero depth reveals that both
the lift-curve slope and the increase in angle of attack due to camber
do change with depth of submersion. At infinite depth the linearized
expression for lift coefficient is given by equation (7). At zero depth
the lift coefficient must be one half of the fully wetted value obtained
from thin-airfoil theory as pointed out in reference 10; that is,


CL,0 = 7t(AO,h + ) (18)


where AO,h and Al,h are the thin-airfoil coefficients of the section
in the hydrofoil plane and are given by the expressions


A, it de (19a)




Alh = g f cos 9 dB (19b)


For the Tulin-Burkart section at zero angle of attack these values may
be determined as





MACA RM L57I16


AO,h = 0.227Ai (20a)


A1,h = 1.151A1 (20b)


Thus, from equations (7) and (18) it is seen that, for a flat plate at
small angles, the lift coefficient goes from i at infinite depth to
a. at zero depth (as given by Green) whereas, for the Tulin-Burkart
section at zero angle of attack these values are .2(l.25Al) at infinite
2
depth and ir(0.802A1) at zero depth. Although the flat-plate lift coef-
ficient doubles in going from infinite to zero depth, the ratio is only
1.28 for the cambered section. The important point to note is that the
value of ac for the Tulin-Burkart section changes from 1.25A, to
0.802Al.

It is now desirable to determine ac for finite depths of submersion.
This can be accomplished by modifying the linearized theory of reference 1
to include the effects of the free water surface.

The effect of the free water surface may be obtained by finding the
transformation which will map the free water surface, the hydrofoil, and
the cavity into the X-axis of the Z-plane. The required transformation
is of the form Z = Z + K1 loge Z + K2. By using this transformation it
is possible, in principle, to determine the forces on the hydrofoil by
following the procedure used in reference 1. However, the solution by
this method became very cumbersome and was abandoned when an approximate
method was discovered. The approximate method continues with the trans-
formation Z = -Z used in reference 1.

In figure 5 it may be seen that Z = -[Z transforms the free water
surface in the hydrofoil plane, where u = 0 (see fig. 5(a)), into a
hyperbola in the third quadrant of the airfoil plane (fig. 5(b)). The
boundary condition that must be satisfied on this hyperbola is that the
perturbation velocity U be zero because in complex velocity problems
of the type considered here, it is the lines of constant velocity which
are being transformed and not lines of constant velocity potential or
stream function.

For the particular case of zero depth the hydrofoil problem is trans-
formed by Z = -fi into the fourth quadrant of the airfoil plane. Thus,
it may be seen in figure 5(b) that the free water surface adds the con-
dition. that u = 0 along the negative Y-axis. This additional boundary
condition can be satisfied (along with the other infinite-depth boundary
conditions) by locating an image of the airfoil-vorticity distribution
along the negative X-axis. The direction of this vorticity must be






16 NACA RM L57E16


opposite to that of the airfoil in order to make d = 0 at all points
along the negative Y-axis. A simpler and often used approximation is to
replace the distributed image vorticity by a single vortex, equal in
strength to the airfoil circulation, at a location equal to the distance
from the leading edge to the airfoil center of pressure as shown in
figure 5(b).

For finite depth of submersion, the condition that u = 0 must be
satisfied at all points on the hyperbola and on the negative X-axis. It
is not possible to satisfy these conditions with a single vortex as was
done for the case of zero depth. However, the influence on the airfoil
of the infinite array of vortices needed to satisfy the boundary condi-
tions shown in figure 6 may be approximated by a single vortex of strength
r in the location shown. The adequacy of the approximation can be deter-
mined by calculating the effect of this image vortex on the lift of a
flat plate as the depth of submersion is varied and then, comparing the
result with the exact solution of Green. This depth effect may be deter-
mined by concentrating the airfoil circulation at its center of pressure
and the image circulation at a point + E forward of the leading
[34 Al
edge and below the leading edge (see fig. 6) and computing the
total downwash on the flat plate at its N point (see ref. 11). The
4
method assumes that the center of pressure of the airfoil remains con-
stant at E/h as the depth changes. The resulting downwash angle a
at the point is calculated to be
4


[ 1 1 + l d4J/c (21)
a 1 (21)
xtV 2 + 2 d/c + (5/4) (d/c)



Since r., = ncaV, the ratio of hydrofoil lift at finite depth CL to the
lift at infinite depth is

CL Cm L r
(22)
CL, C0m,' rL,4 -

therefore


CCL r (23
-c = L =________ (25)
CL,- 1 1 + da/c
1 +2 c + (5/4)1
2 +2jd7c + (/)dc





NACA RM L57I16


Equation (25) is compared in figure 7 with the exact solution of Green
(see fig. 2, a = 0), and the agreement is excellent.

The adequacy of the method used in determining the influence of free
water surface proximity on the lift of a flat plate justifies its use on
cambered foils. However, for cambered foils the problem is more difficult
because a control point such as -L for a flat plate is not sufficient to
4
determine the section coefficients defining the vorticity distribution.
This final vorticity distribution n(s) resulting from the camber and the
image vortex must be determined, particularly if a knowledge of the pres-
sure distribution and thus the drag and center of pressure is desired.

In figure 8, it may be assumed that the final vorticity distribution
on the equivalent airfoil is given by the equation


a(R) = 2V 0 cot + Cn sin n) (24)
\ ~n=1


Then the induced velocity at 2' due to O(R) is, from thin-airfoil
theory (see ref. 12),


v = V -CO + Cn cos no) (25)
\ n=1


The resulting total circulation due to the vorticity given by equation (24)
is, from thin-airfoil theory,


r = rEv(co + (26)


Since the image vortex has a strength equal and opposite to r, the
velocity induced by ri at a point R' on the airfoil is

vi = -- cos 4 (27)
2nrr






NACA RM L57I16


a +1 +-
COS =


a + d + L-
cos AC c

S(a + + )2+ -
Vc ) 4Z c


where


ac=
a =-d-
CL^


CO + Cl -
a = -----
CO +C



Replacing 2' by (l cos 9), equation (28) can be expanded in a
2
Fourier series as

0O
Cos_= f(9) = B0 + Bn cos nG

n= 1


where


B0 1 rf
it- 0


18


where


(28)


(29)


(50)


f(9)de


r =(a + af+ _P2 + 2
C/ VC c 4 4\c





NACA RM L57I16


and


Bn = 2 f(9)cos no dG



By substituting equations (26) and (50) into equation (27) gives


V CO+
vi = 2 BO + Bn cos n (51)
Sn=l



The equivalent airfoil slope x when expanded in a cosine series
is dxR

dja
= -AO + An cos no (52)
n=1


Equating the resulting streamline slope to the equivalent airfoil slope
gives

4 +-~ ^(5
v i
V V d&


The substitution of equations (25), (51), and (52) into equation (55)
gives


02)
-C0 + Cn cos no + Co +-- (BO + Bn cos n = -AO + An cos n9
Z ~ \ 2 L ~
n=1 n=1 n=l
(34)

By equating coefficients of like terms, the C coefficients are deter-
mined as

CO AO( + Bl) + A1BO (35a)
4 + BI 2BO






NACA RM L57116


2A,(2 BO) 2AOB,
Cl = -l -- -.(55b)
4 + B1 2BO


(2AO + AJ)Bn
C, = An (35c)
4 + Bl 2BO


If AO = a and An = 0 where n 1, the effect of depth of sub-
mersion on the lift coefficient of a flat plate can be computed from the
coefficients obtained from equations (55). The values of CL computed
CLWoo
by this method have been found to be in excellent agreement with the solu-
tion of Green for a = 0.

For the particular condition of AO = 0 (the case of hydrofoils such
as the Tulin-Burkart section at zero angle of attack) equations (55)
become


CO = BOA, (36a)
4 + Bi 2BO

S2A,2 BO)
1 4 + B1 2BO


Cn = An BnAl (56c)
4 + BI 2B0


The coefficients BO and Bn as obtained from equation (50) are plotted
in figure 9 against the distance to the center of pressure a for several
depths of submersion. For the special condition of zero depth the B
coefficients for the Tulin-Burkart section are found to be 1.296, 0.772,
and 0.22 for BO, Bl, and B2, respectively. In making this computation,
the final center-of-pressure location a is used; therefore, a is first
given an assumed value, the B and C coefficients determined, then
from the resulting C coefficients, a is calculated from equation (29)
and the procedure repeated if necessary. For the Tulin-Burkart section
a is found to be 0.42 for d/c = 0. By using the final B coefficients
the C coefficients are determined from equations (56) as 0.595Al,
0.646A1, and -0.61Al for CO, Cl, and C2, respectively.






NACA RM L57116


From equation (7) the ratio of the Tulin-Burkart section lift at
zero depth to the lift at infinite depth is

CLI (0.595 + 0.646 + 0.305)Al 1.24
--- = -----~ i ----= 1.26 (57)
CL, B5A1/4


The value 1.24 compares favorably with the more exact value of 1.28 given
in the paragraph following equations (20). The results of calculating
CL
CL for finite depths of submersion for the Tulin-Burkart section are
CL,w
plotted in figure 10. Similar calculations have been made for a circular
arc, and the configurations given in reference 15. The results of these
calculations are also shown in figure 10.

The true linearized lift-curve slope m for finite depths of sub-
mersion in the equation CL = m(a + ac) is that shown in figure 2 for
a. = 0. Therefore, the effective angle of attack due to camber ac is
obtained from the following relationship:


cm0) CL
'(aOj 'c C L
li C
C2 C, L00


Therefore

aCc r CL
Cc,m 2m(a=0) CL,(

a-c
Values of --- are plotted against d/c in figure 11 for the Tulin-
a ,00
Burkart, the circular arc, and the sections given in reference 15.

Equation (58) is obviously limited by the linearizing assumptions
made in its derivation. An important limitation is due to the assumption
that the free surface is always horizontal and thus 5/c = d/c. At
small depth-chord ratios and particularly for large magnitudes of camber
the free water surface is not horizontal and 8/c / d/c. Thus, for small
values of d/c and large magnitudes of camber the values of ac/Cco
given in figure 11 are probably too low.

With a knowledge of the angle of attack due to camber ag at finite
depths of submersion, Green's solution is now modified to include camber






NACA RM L571l6


by treating the effective angle of attack as (a + acg), where ac is
obtained from figure 11. This is exactly the method used in modifying
the Rayleigh equation to obtain the nonlinear approximation for the lift
coefficient at infinite depth. With this assumption, the resultant-force
coefficient for a cambered hydrofoil at any positive depth of submersion
is obtained in terms of the spray thickness 5/c from equations (5) as


CN OJ. CN,ff a +L } (59)


Equation (59) states that the resultant force on a cambered section is
approximated by replacing a in Green's solution for a flat plate by
the effective angle of attack (a + ac). It will be shown that the
resultant force will deviate only slightly from the normal (as previously
pointed out for the condition of infinite depth) and therefore


CLfa. 2 CNffa. + a ccos a (40)


Actually, by itself, equation (40) is of little practical value
because CN,f is given in equations (5) in terms of the usually unknown
parameter 5/c instead of the more useful d/c. Thus, it is necessary
to determine the relationship between 6/c and d/c or at least an
empirical substitute.


Relation Between Depth of Submersion and Spray Thickness

Certain relationships between the leading-edge depth of submersion
and the spray thickness of a flat plate can be stated and are given
as follows:

(1) The trivial but useful case for a. = 0 where d/c = 5/c

(2) The case of a = 900, 6/c = <, where the stagnation streamline
is parallel to the direction of motion and in order to satisfy the con-
tinuity equation 8/e = d/c + 0.5 (see fig. 12(a))

(5) The case of a = 900, 5/c = 0 where again the stagnation line
(this time the free water surface) is parallel to the direction of motion
and 8/c = d/c + 1 (see fig. 12(b))

Cases (2) and (5) suggest that the stagnation line for a = 900 is
also parallel to the direction of motion for finite 8/c (as shown by
the solid line in fig. 12(c)). However, this is not consistent with the





NACA RM L57116


results of Green's analysis where it can be shown from momentum considera-
tions that the streamline is curved as shown by the broken line in fig-
ure 12(c). However, when the effects of gravity and even slight devia-
tions from infinite aspect ratio are considered, the true location of the
stagnation line must lie between these two lines. The tendency of gravity
to cause this change has been discussed already. Possibly more important
than gravity is the influence of finite aspect ratio.

If the supercavitating hydrofoil is replaced by a system of horseshoe
vortices, it can be shown that (for finite angles of attack and infinite
aspect ratio) the stagnation line infinitely forward of the hydrofoil is
also infinitely below the stagnation point as obtained by Green. How-
ever, for an aspect ratio of as much as 100, the stagnation line approaches
a finite asymptote about 0.2 chord below the stagnation point when the
lift coefficient is about 0.5. At a lift coefficient of 0.5, the asymp-
tote becomes about 0.1 chord for moderate aspect ratios. This was observed
in the experiments described in a subsequent section where it was noted
for the angle-of-attack range investigated that the free water surface
was practically undisturbed only slightly forward of the hydrofoil leading
edge. Therefore for an angle of attack of 900 where the total circulation
is zero, it is not unreasonable to expect the solid stagnation line in
figure 12(c) to be more nearly correct than the dashed one when both finite
aspect ratio and gravity are present. It should be noted that although
the stagnation line may deviate from the theory as given by Green, that
the pressure distribution and force on the plate are assumed to remain
unaltered from the expressions given in equations (5). Therefore, if at
900 the stagnation line is assumed parallel to the direction of motion,
it follows that for a = 900


5/c = d/c + 1/c (41)


where 1/c is the dimensionless distance from the leading edge to the
stagnation line (see fig. 12(c)). This distance can be obtained from
Green's work and is, for a = 900

1+ -- + loge( b)- b ( sin-1
i/c = 2 b 1 2 b (42)
1b b2 1) +logb+1
( o e _- )


where b has been previously defined in equations (5).






NACA RM L5716


Case (1) and equations (41) and (42) give the end points (at a = 00
and 900) of the lines of constant d/c which are superimposed on fig-
ure 2. The trends of these curves at intermediate angles of attack were
obtained from experiment and will be discussed in a subsequent section.

The preceding discussion in this report has been concerned with the
effects of gravity and finite aspect ratio on the relationship between the
spray thickness and the leading-edge depth of submersion. It is also of
interest to examine at least qualitatively the effects of gravity and
finite aspect ratio on the spray angle given by equation (4) and plotted
in figure 5. In Green's analysis, it is shown that at a = 900 the total
force on the entire plate is F = pbV2(l cos go; that is, the force is
exactly equal to the change in momentum due only to the flow in the spray.
This means that the stagnation line must be curved as pointed out in the
preceding discussion. The straight stagnation line (which leads to equa-
tion (41)) implies that the change in momentum in the spray flow results
only in the force on that portion of the plate above the stagnation
point. Since the change in momentum of the spray flow now represents
only a portion of the total force on the plate, and since 8/c is con-
sidered to be the same for both stagnation-line curvatures, the angle /
must decrease from the value given by Green in figure 3. That is, the
angle 0 will change so that only the force on the plate above the stag-
nation line is given by the expression F = pBV2(l cos ).


THREE DIMENSIONAL THEORY AT FINITE DEPTH


Lift

The flow about a supercavitating hydrofoil may be constructed by a
suitable combination of sources and vortices. The vortices contribute
unsyimnetrical velocity components and lift, the sources contribute
symmetrical components which provide thickness for the cavity but no
lift. For a finite span the vortices can not end at the tips of the
foil and a system of horseshoe vortices must be combined with the sources
to describe the flow. If it is assumed that the influence of finite span
on the two dimensional lift coefficient is due to the effects of the
trailing vorticity then the resulting effect of aspect ratio is exactly
the same as for a fully wetted airfoil. Jones (ref. 14) gives the lift
of a fully wetted elliptical flat plate as


C
L E 2 (a i)(43)

where E is the ratio of semiperimeter to the span and aj is the
induced angle of attack caused by the trailing vorticity. Thus the effect
of aspect ratio is to decrease the two dimensional lift curve slope by a





NACA RM L5716


factor 1/E and to decrease the effective angle of attack by an incre-
ment o.i. Therefore for the finite aspect ratio supercavitating hydrofoil
at infinite depth, equation (14) is modified to give


CL,1 = L i!( + acL -a) (44)
E 2

or more generally for finite depth, equation (40) becomes


CL,l CNf (a + c "i)cos a (45)

where for rectangular plan form of aspect ratio A, E = A + 1 and
A
C
i = -(A(1 + T) (46)
irA

where T is a correction for plan form (see ref. 12).

Another effect due to finite aspect ratio is the concept of additional
lift due to crossflow (see refs. 15 and 16). This crossflow lift is
assumed due to the drag on the hydrofoil contributed by the component of
free-stream velocity normal to the hydrofoil. In the present case of zero
cavitation number, the crossflow drag coefficient is the Rayleigh value,
0.88. Since this lift is caused only by the spanwise flow (flow around
the ends of the plate) it is also modified to account for the aspect ratio
by the Jones' edge correction, 1/E. Since only the spanwise flow is con-
sidered, E is now the ratio of semiperimeter to chord. Because the flow
being considered is normal to the plate, the induced angle for this flow is
zero. Thus for a flat plate, the crossflow lift CL,c is


CLc 1 0.88 sin2 a cos a (47)


No experimental or theoretical information on the crossflow lift of
cambered surfaces is available in the literature. In order to approximate
this component the following assumptions are made:
(1) The crossflow force acts normal to the hydrofoil chord line.

(2) The effective direction of the free stream on the plate is altered
by the increase in angle of attack due to camber ac.

Thus, the crossflow lift on cambered sections is assumed to be





NACA RM L57I16


CL n ]1 0.88 sin2(a, + cLcos a' (48)
CL, A q 1 8 c

where a' = a + %o, and so is the inclination of the chord line to the
reference line of the section (positive if the chord line is below the
reference line) and ar is obtained from figure 11 for the depth of
interest.

The total lift on a finite aspect ratio hydrofoil operating near the
free water surface is therefore obtained by adding equation (48) to equa-
tion (45) to give


CLJal A A Cn f{m + ac ai cos a + A 1 0.88 sin2 I' + acLc)cos a'
"[j A + 1 n' 'IA + 1 v c/
(49)


In view of the very approximate nature of equation (48) it is desir-
able to examine the effect of this crossflow term on the total-lift coef-
ficient. For the Tulin-Burkart, aspect-ratio-1 section (Al = 0.2)
used in this investigation, at d/c = 0.071 the ratio of the calcu-
lated crossflow lift, CL,c to the calculated total lift was 0.157
at a = 40 and 0.285 at a = 200. For the five-term section of refer-
ence 15, where Al is 0.075, the aspect ratio is 5, and d/c is 0.071,
the ratio has been calculated as 0.014 at a = 40 and 0.072 at a = 200.
Thus any inaccuracies in the crossflow lift as computed by equation (48)
will appreciably affect the total-lift coefficient at large angles, small
aspect ratios, and large cambers. On the other hand at higher aspect
ratios and small cambers, errors in the crossflow component do not greatly
influence the total lift.

Equation (49) may be written in terms of the slope m (given in
fig. 2) as


A cosa +
S( C L + 0c i) COS -- +
A + 1 cos(a. + % a)


1 0.88 sin2(a' + m c)cos a' (50)
A + 1

where mc is obtained from figure 11 for the depth-chord ratio of
interest and am is obtained from equation (46). In equation (46),
CL i is the first term in equation (50). Equation (50) is solved by
iteration and the convergence is quite rapid. A sample calculation of
the lift coefficient of a cambered hydrofoil is given in appendix B.





NACA RM L57116 27


Drag

The drag coefficient of a supercavitating hydrofoil of finite aspect
ratio operating at zero cavitation number and finite depth of submersion
is


CD = CL ,l tan(a. + E) + CLc tan a' + C (51)


where CL,1 is the first term in equation (50O) and c is the deviation
of the resultant-force vector from the normal. For a flat plate e = 0,
a.' = a, and thus CD = CL tan a + Cf. For cambered surfaces similar to
the circular-arc or Tulin-Burkart section, e becomes very small at large
angles of attack and may be neglected; however, at small angles of attack,
the effect of E on the drag coefficient cannot be neglected. An approxi-
mation to the value of c can be made by determining its value from the
two-dimensional linearized solution and then modifying the result for the
case of finite angles of attack and aspect ratio.

The linearized drag coefficient as given in reference 1 is



CD = R di (52)
10 cV2


replacing by T(1 cos 9) and di by -' sin 9 d9 equation (52)
becomes



CD f 1 cos 8)sin 6 71 d (55)
0 r V2


Now 7 may be written in terms of the vorticity on the equivalent air-
foil operating at finite depth as



= V CO + -Cos + Cn sin nG (54)
n=1





NACA RM L5716


Sosn00
S= V -AO + x An cos no)
n=l


(55)


Therefore


CD = (1 cos 9)sin 9 Co1 si+ cos 9) + C sin no (-AO + An cos n)d9

(56)
For the condition where A0 = aC, CD becomes after integrating


CD = (Cg 0 + Cl


(- )A CO +C(c A A2


(57)


At infinite depth (CO = A0, Cn = An) equation (57) reduces to the
value given by Tulin-Burkart in reference 1.


=CD, (A + Al 2
CD)m 2


(58)


For AO = a, that is A0' = 0


SCD
CL


M (C + Cl 2


Al CO + C21+2A

Cl- C2)2

(59)


Therefore


S(C1 2C2)Al + (2CO + 2C1 + C4)A2 A4C2
S(co + Cl )


(60)


The value of e given by equation (60) is adequate only for the case of
small angle of attack and camber and depth-chord ratios larger than


+ a Cl
+2 G_


-C +
2 ) 2


- 2
87


(CO +






NACA RM L57I16


about 1. At small spray thickness to chord ratios it may be seen in
figure 15 that the spray angle becomes quite large even for small angles
of attack. When such a flow is transformed by Z = -%f the cavity
streamline and the free water surface are rotated as shown in figure 15.
In figure 15(b) the boundary conditions are now different from the simple
u = 0 used in the small-angle theory. If a system of vortices could be
located to satisfy the boundary conditions along these new lines, a solu-
tion for the resulting vorticity on the foil could be obtained. Such a
method would involve taking a different spray angle (from fig. 5) for
each depth and angle of attack in order to locate the image vortex or
array of vortices. Also, for large angles of attack and spray angles
the linearizing assumption of 7 << V is not adequate. In calculating
the lift, these difficulties were avoided by using Green's solution which
takes the effect of the spray angle into account. It can be seen in
figure 2 that as the angle of attack increases, the ratio of dimin-
CL,
ishes, therefore, the image vortex must have less influence on the
resulting hydrofoil vorticity for large angles of attack. An approxima-
tion to the correct hydrofoil-vorticity distribution for finite angles
of attack operating near the free water surface can be obtained by using
the model shown in figure 8, and increasing the value of d/c used so
that the resulting CCL- corresponds to that given by Green. Thus, for
0L,-
large angles of attack, camber, and finite aspect ratio an effective
depth of submersion, (d/c)e should be used in equation (60). The value
of (d/c)e is the value of d/c on the a = 0 line in figure 2 corre-
sponding to the value of m = me, where



CL x m@a + Go a1} xt/-
Lme v m-+ac q 'L A (61)
CL,o 2 {a -c ai) m 2


The value of the C coefficients are then determined for (d/c)e and
AO = a CLa .

A sample calculation of the drag coefficient is given in appendix B.
It has been found after several calculations that the value of e and
thus the drag coefficient is not greatly affected by the depth of sub-
mersion. In fact, a rough approximation to E may be obtained by assuming
CO = a a1, and Cn = An in equation (60).






NACA RM L57116


Center of Pressure

The linearized expression for the center of pressure of a finite-
aspect-ratio, supercavitating hydrofoil operating at zero cavitation num-
ber and finite depth of submersion is
Ci.
-m 1_ 1 +0 7Cl TC2 + 3C C
x 1= -, O 7l--C2 --- (62)
cp CL,l 16 o 2


where the C coefficients are determined at the effective depth of sub-
mersion given by equation (61) and for AO = a ac. Superimposed on
this flow is the crossflow component of lift which is assumed to be
distributed uniformly over the chord and acting in a direction normal to
the chord line. Thus, the distance from the leading edge to the center
of pressure of the crossflow-lift component xc.p.,c is given by


Xc.p.,c = 0.5c (63)


Admittedly, this assumption is crude and accurate only for a flat plate.
For cambered surfaces the crossflow will not be uniformly distributed
and for low drag cambered sections is probably concentrated on the rear-
ward portion of the hydrofoil.

By combining equations (62) and (65) the center of pressure of the
combined flows is, therefore,

(CL lxc.p., 1 + 0.5CLc (6)
CL 0(4

A sample calculation of the center of pressure is given in appendix B.
As in the case of e, a few calculations reveal that a fair approximation
for Xc.p.,l is obtained by using CO = a ai and Cn = An in equa-
tion (62).


EXPERIMENTAL INVESTIGATION


MODELS


Two models of 7.071-inch chord were used in the experimental inves-
tigation. As shown in figure 14(a), the first had a lower surface





NACA RM L57116


conforming to the Tulin-Burkart low-drag configuration given by equa-
tion (12). The two-dimensional design-lift coefficient was selected as
0.592 corresponding to a value of 0.2 for the coefficient A1. Since the
hydrofoil is designed to operate in a cavity, the shape of the upper sur-
face is arbitrary as long as it does not interfere with the formation of
the cavity from the leading edge. Since the greatest advantage is to be
obtained at small angles of attack and thus thin cavities, the thickness
of the hydrofoil must be small. For the present cambered model the upper
surface profile from leading edge to midchord was arbitrarily chosen to
conform with the free streamline leaving the leading edge of a two-
dimensional flat plate at 50 incidence. (See ref. 1.) The thickness of
the portion rearward of the midchord was made in the image of the forward
portion and resulted in a symmetrical thickness distribution with a maxi-
mum thickness-chord ratio of 53.5 percent. Since the center of pressure
of the Tulin-Burkart hydrofoil is located near the midchord, this symmet-
rical section minimizes the torsional moment on the foil and results in
less twist than would be experienced by a nonsymmetrical section.

The second model was of triangular cross section with a flat bottom
as shown in figure l1(b). The maximum thickness was 5 percent of the
chord.

Such thin sections lead to structural limitations in the aspect ratio
when supported by a single strut at midspan. Since an aspect ratio of 1
is about the most desirable from the structural standpoint and it also
represents the accepted dividing line between hydrofoils and hydroskis,
both models were made with a square plan form.

The strut, which can also be seen in figure 14, had an NACA 661-012
airfoil section. The strut was mounted perpendicular to the flat plate
and perpendicular to the X-axis of the cambered surface. The intersection
of the strut and upper surface was without fillets. Both the hydrofoil
and the strut were made of stainless steel and were polished to a smooth
finish.


APPARATUS AND PROCEDURE


Tests were made by using the carriage and existing strain-gage bal-
ances in the Langley tank no. 2 which independently measure the lift,
drag, and pitching moment. Figure 15 shows a view of the test setup with
the cambered hydrofoil and the balance attached to the carriage. The
moment was measured about an arbitrary point above the model and the data
thus obtained were used to calculate the moments about the leading edge.
The positive directions of forces, angles, and moments used in presenting
the force data are shown in figure 16.






NACA RM L57116


The force and moment measurements were made at constant speeds for
fixed angles of attack and depths of submersion. The depth of submersion
is defined as the distance from the undisturbed water surface to the
leading edge of the model.

Ventilated or zero-cavitation-number tests were made on both models
at a depth of submersion of 0.5 inch throughout a range of angle of attack
from 60 to 200 for the flat plate, and 80 to 200 for the cambered hydro-
foil. Three methods of obtaining ventilated flow at this 0.5-inch depth
were used: (1) Normal ventilation through the trailing vortices as
described in reference 4, (2) injection of air through the port on the
strut leading edge (see fig. 14) (this air was supplied at a rate of
0.012 pounds per second and cut off after ventilation was established),
and (5) a 1/52-inch-diameter wire was soldered on the leading edge of
the cambered model to cause local separation and thus ventilation. Both
models were also investigated at a depth of submersion of 0 inch at
a = 40 for the flat plate, and a = 60, 80, and 100 for the cambered
surface. At angles of attack of 160 and 200 for the flat surface, forces
were also measured over a range of depth of submersion for which venti-
lation could be obtained (d = 0 to 2 inches).

The thickness and direction of the jet or spray leaving the leading
edge of the flat plate were also measured at 160 and 200 for a range of
d from 0 to 2 inches. A schematic drawing of the instrument used for
measuring the spray thickness and direction is shown in figure 17. The
stagnation tube was lowered through the spray during a test and the pres-
sure and location of the tube center line recorded on an oscillograph.
Almost instantaneous response of the stagnation-tube-pressure-cell
combination was obtained by completely filling the tube and connecting
line with water. The point of entering and leaving the spray was obtained
by comparing the location of the tube with the rise and fall of pressure
as the tube passed through the spray. The vertical location of the tube
was obtained from the output of the slide-wire circuit also shown in fig-
ure 8. During a test the tube was passed through the spray several times
and an average of the results was taken.

Tests of the nonzero-cavitation-number case were made at a depth of
submersion of 6 inches where vortex ventilation did not occur. Measure-
ments of lift, drag, and moment were obtained for a range of velocities
from 20 to 80 feet per second at angles from 160 to 200 for which long
trailing air or vapor cavities could be obtained. Data for air-filled
cavities were obtained by introducing air from an external metered supply
to the upper surface through the ports on the strut leading edge shown
in figure 14. During these tests the pressure in the cavity formed was
measured by a pressure cell connected to a 1/16-inch-diameter orifice
located near the bottom of the strut so as to be within the cavity. (See
fig. 1k.) This measured pressure was used in computing the cavitation





NACA RM L571l6 55


number for the cavity formed. The airflow rate was measured by an
orifice-type flow meter. Airflow rates up to 0.012 pound per second
were obtained with the test arrangement.


ACCURACY


The change in angle of attack due to structural deflection caused
by the forces on the model was obtained during the calibration of the
balances and test data were adjusted accordingly. The maximum correction
necessary was only 0.10. The estimated accuracy of the measurements is
as follows:

Angle of attack, deg . . 0.1
Depth of submersion, in. . . 0.1
Speed, fps . . . 0.2
Lift, Ib . . . 15
Drag, lb . . . 7
Moment, ft-lb . . . 6
Cavity pressure, Ib/sq ft . . 10
Spray thickness, in. . . 0.05
Spray angle, deg . . . 1.5

The forces and moments were converted to the usual aerodynamic coef-
ficient form by using a measured value of the density of 1.95 slugs per
cubic foot. The kinematic viscosity measured during the tests was
1.70 x 10 pounds-second per square foot. Thus, for the range of veloc-
ities investigated, the Reynolds number based on chord ranged from
0.7 x 106 to 2.8 x 106.


RESULTS AND DISCUSSION


ZERO CAVITATION NUMBER


General

Ventilation inception.- The process of ventilation by air entrainment
through the tip vortices of a hydrofoil near the free water surface is
described in reference 4 and is shown for the present models operating at
a depth of 0.5 inch, in figures 18 and 19. The speed at which complete
ventilation occurred, when tested at this depth is shown in figure 20.
Complete ventilation by entrainment through the vortices was not possible
at angles less than 100 for the flat surface and 120 for the cambered





NACA RM L57I16


surface. Figure 18(a) shows the incomplete ventilation of the flat plate
at So and figure 19(a) the similar action of the cambered surface at 100.
Figure 18(b) shows the complete ventilation of the flat surface at 100 and
figures 19(b) and (c), the complete ventilation of the cambered surface
at 120 and 160, respectively. Leading-edge cavitation of both models may
be noted in the photographs.

One of the requirements for establishing ventilated flow from the
leading edge is that the hydrofoil upper surface lie beneath the upper
cavity streamline for the condition of zero cavitation number. Thus,
for a given hydrofoil thickness there is a minimum angle below which
ventilation from the leading edge cannot exist. Figure 21 shows the
theoretical infinite depth location of the upper cavity boundary for two-
dimensional hydrofoils having the same cross section as the models tested.
These streamlines, computed from equations given in references 1 and 15,
reveal that ventilation of the two-dimensional flat plate should be
possible at angles greater than 3.20

As pointed out in references 1 and 15, the slope of the cavity
streamline is determined by the magnitude and distribution of the circu-
lation on the equivalent airfoil section. The reduction of the two-
dimensional (infinite depth) circulation by an amount equivalent to
Jones' edge correction, 1/E = A/(A + 1), will cause the cavity streamline
to be lowered over the entire span by an amount proportional to 1/E.
When the induced angle ac is considered, the streamline is brought even
closer to the hydrofoil upper surface, particularly near the tips.

By considering these two effects and by assuming that the influence
of am is uniform over the span, the effect of finite aspect ratio on
the cavity slope ( a) may be approximated by the following equation:


fdya ai)
_c -A+ 1 (65)

[/dy\j 0() A--
dx A=-o

Thus, in order to establish a cavity at angle of attack a and aspect
ratio A which is equivalent to the cavity formed at angle of attack a
and A = w the following equation must be satisfied:


A + (0)A=' + ai. (66)






NACA RM L57I16


By using equation (66) and the value of (cL)A= = 5.20, the geometric
angle of attack of the aspect-ratio-1 plate would have to be about 80
in order to ventilate from the leading edge at infinite depth of sub-
mersion. The effect of the proximity of the free water surface is to
rotate the cavity streamline away from the hydrofoil as seen in fig-
ure 5. Thus, it should be possible to establish a vented cavity on the
flat section at angles at least as small as 80 when operating near the
free water surface. Therefore, the absence of ventilation at angles
less than 100 cannot be attributed to interference from the model upper
surface. In fact the formation of leading-edge cavitation at o' indi-
cates that if the velocity could have been increased well above 80 feet
per second (preferably to zero cavitation number) a vapor cavity com-
pletely inclosing the upper surface would result. The reason that venti-
lation did not occur at angles less than 100 could be that the region of
boundary-layer separation in the midportion of the chord did not extend
to the aerated vortices. This conjecture was proved to be the case by
injecting air from the external supply down the strut to the upper sur-
face of the model. Complete ventilation of the upper surface was then
possible at angles as low as 60. The air was supplied (at the maximum
rate of 0.012 pound per second) during the acceleration of the model
and shut off after ventilation was established. The process is shown
in figure 18(c) for the flat plate at 80 angle of attack.

Figure 21 shows that ventilation of the two-dimensional cambered
section should be possible at angles above about 40. In reference 15
at large angles of attack, the cavity shape is almost independent of the
camber, depending only on the angle of attack. Therefore, the finite-
aspect-ratio corrections given by equations (65) and (66) are assumed
applicable to cambered sections. When these corrections for finite
aspect ratio are taken into account the minimum angle for ventilation of
the cambered section becomes about 120. Again the proximity of the free
water surface will reduce this infinite-depth estimate. Thus, failure
to establish complete ventilation of this hydrofoil at angles less than
120 (at d = 0.5 inch) may have been due to interference of the upper
boundary of the model. In fact it was found that injecting air to the
upper surface at angles less than 120 failed to establish ventilation
or to influence the ventilation speed at 120. Since the lift-drag ratio
of cambered supercavitating hydrofoils increases with decrease in angle
of attack, these observations suggest the use of higher geometric aspect
ratios or the effective increase of aspect ratio by the use of end plates.
End plates appear to be particularly applicable if low-aspect-ratio
cambered hydrofoils are to be designed to ventilate from the leading edge
at small angles of attack and large depths of submersion.

Since complete ventilation seemed to require extensive boundary-
layer separation on the upper surface, a 1/52-inch diameter wire was
soldered on the leading edge of the cambered model to cause a local
boundary-layer separation. With this wire in place it was possible at






NACA RM L57116


the 0.5-inch depth of submersion to obtain complete vortex ventilation
of the upper surface at 100 incidence but not at 80 incidence even at the
maximum available velocity of 80 feet per second. However, by introducing
air through the ports on the strut it was possible to obtain complete
ventilation at angles as low as 80. Evidently air brought up by the tip
vortices at 8 incidence could not reach the small separated region behind
the wire; however, the forced air was capable of reaching this zone and
establishing a cavity. The cavity formed extended downstream far enough
to intercept the ventilated tip vortices and complete ventilation then
occurred. Of course, the primary influence of the wire was to increase
the cavity ordinates so that the hydrofoil upper surface did not inter-
fere with establishing ventilation from the wire. Although the basic
force data obtained with the wire were recorded and are presented, they
are not used in the later correlation with theory because of the unknown
influence of the wire on the forces.

Basic force and moment results.- The basic data from the tests of
the ventilated hydrofoils at a depth of submersion of 0.5 inch are pre-
sented in figure 22 for the flat plate and figure 25 for the cambered
surface as curves of lift, drag, and pitching moment about the leading
edge against speed, for various angles of attack. Ventilated-flow data
obtained at zero depth of submersion at 40 incidence for the flat plate
and 60, 80, and 100 for the cambered surface are also included.

The basic ventilated flow data obtained for depths of 0, 1.0, 1.5,
and 2.0 inches for incidences of 160 and 200 are presented in figure 24.


Comparison of Experimental Results With Theory

Spray thickness.- In the section on theory the need for determining
the relationship between the leading-edge depth of submersion and the
spray thickness for a flat plate was pointed out. These variables were
measured for the flat plate at 160 and 200 over the range of depth of
submersion from 0 to 2 inches. The data are shown in figure 25. The
spray-thickness measurements presented were obtained approximately 20 per-
cent of the chord rearward of the leading edge. On one test, measurement
of the spray thickness was also obtained at about the midchord and the
results were in agreement with those obtained at the 20-percent location.
Therefore, for the range of depths investigated, the spray thickness
measured can be considered as the theoretical value infinitely rearward
of the foil. It may be noted in figure 25 that the spray thickness is
greater than the leading-edge depth of submersion, the magnitude of the
ratio increases with either increase in angle of attack or decrease in
depth of submersion. Also shown in figure 25 are the theoretical rela-
tionships between 8/c and d/c for the two-dimensional 00 and 900 cases
previously discussed. It may be noted that the trends of the experimental
and theoretical curves are parallel.






NACA RM L57116 57


In foregoing sections it has been pointed out that the relationship
between leading-edge depth of submersion and spray thickness is influ-
enced by the force of gravity and thus cannot be obtained from Green's
analysis. Also the influence of aspect ratio has been shown to be impor-
tant. The relationship for a = 00 and 900 has been determined but the
manner in which intermediate angles of attack affect the result was not
understood. The data shown in figure 25 combined with the end-point
results previously established for a = 00 and 900 permit lines of con-
stant d/c, at least for the aspect-ratio-1 condition, to be drawn on
figure 2. The experimental data shown in figure 25 were plotted on fig-
ure 2 at their equivalent angle of attack a mi. This corresponds to
equivalent angles of attack of approximately 12.50 for the 160 case and
15.90 for the 200 case for the range of d/c presented. For values of
d/c greater than 0.285, only the theoretical end points were available
and paralleling lines were faired in. Although these lines are accurate
only for the aspect-ratio-1 condition, they are considered to be good
approximations even for aspect ratios as high as 6. Such an approxima-
tion is reasonable because the asymptotic value of the stagnation line
infinitely forward of a flat plate is only of the order of about 0.1 chord
below the stagnation point for aspect ratios less than 6 and lift coef-
ficients less than 0.5. It is obvious from figure 2 that for depths
greater than 1 chord, the end points may be connected by any reasonable
line (for example, a straight line) with very little loss in accuracy.

Spray angle.- Figure 26 shows the effect of depth of submersion on
the spray angle $ for the flat plate at 160 and 200. The theoretical
spray-angle variation also shown in figure 26 was obtained from figure 5
by considering the two-dimensional angle of attack as a aO and the
two-dimensional 5/c as the actual measured value. The tangent of the
spray angles obtained from figure 5 for the angle a aCL were then
reduced by the factor 1/E. The justification for this 1/E modification
is the same as that presented in the section on ventilation inception.
That is, that the slope of the upper cavity streamline is proportional
to the equivalent airfoil circulation and thus if the two-dimensional
circulation is reduced by 1/E, the two-dimensional spray-angle slope
will also be reduced by this amount. When these three-dimensional cor-
rections are made, the calculated angles are still high at small values
of d/c and appear to coincide with the experimental data for values
of d/c greater than about 0.25. Since the theorem that the slope of
the cavity streamline is directly proportional to the circulation was
obtained in reference 1 for very thin cavities at infinite depth; it
cannot be expected to be applicable when the spray angles are very large.
This deviation from assumed small angles may be the only explanation
required for the discrepancy between theory and experiment at small values
of d/c. However, it might also be attributed to the influence of gravity
and finite aspect ratio on Green's solution. In the section entitled






NACA RM L57TI16


"Relation Between Depth of Submersion and Spray Thickness," it was sug-
gested that the spray angles given in figure 3 would be reduced because
of these effects and at least this assumption has not been disproved by
the experimental data.

Lift coefficient.- All ventilated force and moment data in coefficient
form were found to be independent of speed in the range tested. (This
independence was true in the present investigation because of the shallow
depth of submersion and therefore a 0. At very large depths, a will
be greater than zero because po is greater than p0 even if the cavity
is fully vented to the atmosphere. Therefore at large depths, changes in
velocity will affect the lift coefficient because these changes affect a.)
The data shown in figures 22(a) and 25(a) are plotted in figure 27 as lift
coefficient against angle of attack for each of the models tested. A
comparison of the lift coefficients of the two models shows an effective
increase in angle of attack of the cambered model as predicted by the
Tulin-Burkart theory. Also shown in figure 27 are the theoretical lift-
coefficient curves obtained from equation (50). The theory is about
5 percent lower than the measured values. Thus, the use of equation (50)
in engineering calculations of the lift coefficient appears to be warranted.

The variation of lift coefficient with depth of submersion of the
flat-plate model at angles of incidence of 160 and 200 is shown in fig-
ure 28. Note the slight increase in lift coefficient as the hydrofoil
nears the surface. Also shown in figure 28 is the theoretical variation
of the lift with depth of submersion obtained from equation (50). The
theory is in excellent agreement with the data and accurately predicts
the increase in lift with decreasing depth of submersion.

Further verification of equation (50) is shown in figure 29 where a
comparison is made with the experimental data of Fuller (ref. 17). These
data were obtained on sections with lower surface profiles similar to
those of the present investigation but at an aspect ratio of 2. The
calculated values are in good agreement with the measured values for
both the flat and cambered models.

Drag coefficient.- In figure 30 the data of figures 22(b) and 23(b)
in coefficient form are compared with theoretical values obtained from
equation (51). The friction drag coefficient of one side of either of
the models was calculated to be about 0.005. With the strut drag included,
the total-skin-friction drag coefficient Cf was taken as 0.004. By
using this value of Cf in equation (51), the agreement between theory
and experiment is good for both models.

In figure 51 the experimental lift-drag ratios obtained from the
data of figures 22 and 25 are compared with theory. Again, both experi-
ment and theory include the skin-friction drag coefficient (Cf = 0.004).





NACA RM L5716


The agreement between theory and experiment is good for both models.
The superiority of the cambered hydrofoil is clearly revealed in this
figure. At a lift coefficient of 0.25 the L/D of the cambered hydro-
foil is more than twice that of the flat plate.

Also included in figure 51 are data taken from reference 4 on a
ventilated modified flat plate of aspect ratio 1. This modified plate
had an elliptical nose and a tapered trailing edge. The importance of
providing a sharp leading edge on hydrofoils designed for use in cavity
flow is shown by comparing the L/D of this modified flat plate with
the L/D of the sharp-nosed flat plate of the present investigation.
The rounded leading edge of the modified plate is subjected to a net
positive pressure which is not balanced by similar pressures on the rear-
ward portion of the plate. In addition, the lower surface of this plate
presents an effective negative camber to the flow and thus does not develop
as much lift as a truly flat surface. At small angles the drag of the
rounded nose greatly influences the maximum L/D of the section. Several
other investigations have noted the importance of a sharp leading edge on
hydrofoils designed for operation in the supercavitating regime. (See
refs. 18 and 19.)

Center of pressure.- The center of pressure of the flat and cambered
models calculated from the data of figures 22 and 25 are compared with
theory in figure 52. The theory from equation (64) is in good agreement
with the experimental data for both models. Since the accuracy of the
forces and moments on the flat plate is poor at small angles of attack
(small total loads) the accuracy of the center of pressure from the data
obtained on the flat plate at angles of incidence of 40 and 60 is doubtful.


NONZERO CAVITATION NUMBER


The nonzero-cavitation-number characteristics of the two models
obtained at a 6-inch depth of submersion are shown in figure 3355 for angles
of attack of 160, l80, and 200. The solid data points are the lift coef-
ficients obtained for vapor cavitation. The cavitation number corresponding
to the condition tested was computed by using the water vapor pressure at
the test temperature for the pressure within the cavity (that is
a = P q Pv). For cavitation numbers less than about 0.7 the vapor pres-
sure was the same as the measured cavity pressure. However, at cavitation
numbers greater than 0.7, the measured pressure was usually higher than
the vapor pressure. Since it could not be determined with certainty
whether the cavity pressure orifice was within the cavity, the vapor pres-
sure was used to compute all cavitation numbers denoted by the solid data
points. Also denoted in figure 3355 are the approximate values of a at
inception ai and the point at which the cavity length exceeded the chord.






NACA RM L5716


These values are only estimates, since no effort was made to find the
exact velocity at which these incidents occurred. Although the estimated
values of ai are the same for both models, there may actually be some
difference in the true points.

The blank data points in figure 55 represent data obtained by intro-
ducing air to the upper surface of the model and establishing a cavity.
The cavitation number for this condition was computed by using the
measured value of the pressure within the cavity. In this case the cavity
pressure orifice was always well within the cavity formed.

In figure 55 the curves are drawn through the computed value of CL
for a = 0 as obtained from equation (50).

The agreement between the vapor- and air-cavity data confirms the
use of the cavitation number as the significant parameter for correlating
the characteristics of cavity flow. The similarity of the air and vapor
cavities at nearly equal cavitation number is shown in figure 54. This
dependence on the cavitation number is to be expected because the forces
on the body are influenced only by the streamline curvatures and thus the
pressure within the cavity; the type of gas present should have only a
secondary influence.

It was not possible to establish satisfactory air cavities at angles
less than about 160. At low speeds where the air could reach the sepa-
rated region near the leading edge the cavity upper surface was greatly
disturbed by the force of the air jets. At higher velocities either the
air could not reach far enough upstream to form a cavity or when a cavity
was formed it did not cover the whole chord. If a greater quantity of
air is supplied it is believed that satisfactory results can be obtained
at angles less than 160.

The use of a dynamic model of a high-speed aircraft will require
simultaneous reproduction of both the full-scale cavitation and Froude
numbers. Such an investigation is possible if an air-filled cavity with
the proper cavitation number can be established on the model.


CONCLUSIONS


Conclusions based on the results obtained from the theoretical and
experimental investigations of supercavitating flat and cambered hydro-
foils may be summarized as follows:

1. The theoretical expressions derived for the lift, drag, and
center of pressure of supercavitating hydrofoils of arbitrary section





NACA RM L57116


operating at zero cavitation number, and finite aspect ratio and depth
of submersion are in good enough agreement with the available experimental
data to warrant their use in engineering calculations.

2. The experimental ventilated force and moment coefficients of both
models investigated at shallow depths of submersion were independent of
speed.

5. Similar to the influence of camber on a fully wetted airfoil, the
influence of camber in ventilated flow was to effectively increase the
angle of attack without appreciably changing the lift-curve slope.

4. The lift-drag ratio of the cambered model near the design lift
coefficient was more than twice that of the flat plate.

5. Comparison of the sharp-nosed flat plate with a rounded-nose flat
plate showed the sharp-nosed section to be considerably superior. Thus,
hydrofoils designed for operation in the supercavitating regime of flow
should have sharp leading edges for best efficiency.

6. The cavitation number defines the flow similarity and lift of a
supercavitating hydrofoil regardless of the type of gas (air or water
vapor) within the cavity. Thus vapor cavities, normally obtained at high
speeds can be simulated at lower speeds by establishing a cavity with
forced air whose cavitation number is the same as that of the high-speed
flow.


Langley Aeronautical Laboratory,
National Advisory Committee for Aeronautics,
Langley Field, Va., August 27, 1957.






42 NACA RM L57116


APPENDIX A


THE TULIN-BURKART LINEAR THEORY FOR ZERO-CAVITATION-NUMBER

FLOW APPLIED TO A CIRCULAR-ARC HYDROFOIL


The problem is to find the effective increase in angle of attack due
to the camber of a two-dimensional circular-arc hydrofoil operating at
infinite depth and zero cavitation number by the method of reference 1,
that is, to find the section coefficients AO, Al, and A2 in the fol-
lowing equation if a, measured from the chord line, is zero:


CL = (AO' + A )= (AO +ajc)


By using the system of axes shown in the following sketch


and the notation of reference 1 (barred symbols for the equivalent air-
foil and unbarred for the hydrofoil), the equation of the circular-arc
hydrofoil is


(x- c/2)2 + (y + R cos 7/2)2 = R2


(Al)



(A2)


dy x -c/2
-x R2 (x c/2)2





NACA RM L57116


From reference 1 the slope of the equivalent airfoil is obtained from
the equation


di


- x
dx


(A5)


By noting that c = F2, the slope of the equivalent airfoil section is,
therefore,


dy 2 8-2/2-
d R2 (2 52/2)2


If the conventional substitution of

S= E(1 cos 9)
2

is made, equation (4) after some manipulation may be written as


S(1 cos 9)2 1
dy T


(Au)


(A5)


(A6)


- (i cos )2 2
1 4


Therefore,


d 1
di 2


R2
C2
7-


R 2 1-2
(R2 sin
2


(A7)


2
cos 9 2 cos 9 1

/cos2 2 cos 9 1
rsin2 Z 2 -
2


(A8)


d ~






NACA RM L57I16


Since the linear theory is applicable only to small
therefore,


7 and,


1 cos20
sin2 Z
2


- 2 cos 9 -
2


equation (A8) may be approximated as


St (cos2 2 cos 9 1)
dxi 4


The necessary coefficients are then readily obtainable as


A = = -d Z
In dia3 8



A, 2 4f cos 6 d =Z
A J dx 2


A2 1( f
x ,


0 cos
d3


An = 0


The required effective increase in angle
arc camber is, therefore,


29 d9 = -
8


(n > 2)


of attack due to small circular-


(A14)


A- A2 1f -= 9
2 2 2 8) 167


It should be noted that for the reference line used in the analysis,
AO' is not zero and positive lower surface pressures cannot possibly be
realized near the leading edge unless the angle of attack is increased at


(A9)


(Alo0)




(All)


(A12)




(A15)






NACA RM L57I16


least to the point where
Al sin 9 + A2 sin 29 is
the condition a 2 = 0
8
over the entire chord of


a = 0. Since An = 0 (with n > 2) and
everywhere positive in the interval 0 5 9 5 v,
is sufficient to specify positive pressures
the hydrofoil.


A convenient way of treating the circular-arc section to make it
comparable to other low-drag sections is to reorient its reference line
so that ao = that is, AO' = 0. This orientation then corresponds
to a = 0. At this design angle of attack a = 0, ao = the hydro-
dynamic efficiency CL/CD of a two-dimensional circular-arc section as
computed from linear theory is 1 -I-, which is almost as good as the
16 2CL L 0
low-drag hydrofoil selected in reference 1, ICL 7- .
-C i6 2CL /






NACA RM L5716


APPENDIX B


SAMPLE CALCULATION OF LIFT, DRAG, AND CENTER OF PRESSURE

OF A CAMBERED LIFTING SURFACE OPERATING AT FINITE

DEPTH AND ZERO CAVITATION NUMBER


A sample calculation of the lift, drag, and center of pressure is
presented for a Tulin-Burkart section with Al = 0.2(CLd = 0.592),
having an aspect ratio of 1 and operating at an angle of attack of 120
and depth of submersion of 0.071 chord.


LIFT COEFFICIENT


Step 1

For the Tulin-Burkart hydrofoil section
tion (7),


(A2 =-


2)AI from qua-
]-= from equa-


2 4A
ag,. = Al ][" ~- = 0.25



Step 2

From figure 11 at d/c = 0.071,


-= 0.718
7ac,-_


Therefore,


ac = (0.718)(0.25) = 0.18





NACA RM L57116


Step 5


By assuming that
tion (1+6),


CL,l = 0.25 and T = 0.12 and by using equa-


a- = 1.12 0-2 = 0.088
Xt


and


a + ac a 123 + 0.18 0.088 = 0.301 radian = 17.20
57.5


From figure 2 for d/c = 0.071 and a. = 17.20, it is found that m = 1.63;
therefore, from reference 11, T = 0.12 as assumed. By using equa-
tion (50),

I cos 12
CL1 = .(1.65)(0o.30o1)co 120 0.251
L 2 cos 17.20

This should check with the original assumption if not, repeat step 5
with better approximation for CL and T.


Step 4

For the Tulin-Burkart section given by equation (12) with Al = 0.2,


a(1)


-Al
6


Therefore,


o0 = tan-10.033 = 1.920




a' = 120 + 1.920 = 153.920

a = 0.18 radian = 10.30






48 NACA RM L57116


or

a, + ac = 24.220


By using equation (48),


CLc = 0.88 sin2(24.22)cos 15.920 = 0.072


Thus, the required lift coefficient is


CL = CL,1 + CL,c = 0.251 + 0.072 = 0.525



DRAG COEFFICIENT


Step 1

From step 5 in the lift-coefficient calculation,


a. + ac am = 17.20


For d/c = 0.071, m = 1.65; and for d/c = a0, m = 1.2. Therefore, from
equation (61)


me = L63 2.15
e 1. 2 2

From figure 2 by using the a. = 0 line with m = 2.15, it can be found
that

(d/c)e = 0.66





NACA RM L57I16


Step 2

If a (the distance to airfoil center of pressure) is assumed to
be 0.57, then from figure 9 for a = 0.57 and d/c = 0.66, the following
values are found:


BO = 0.56


BI = 0.15

B2 0


From equation (35) for
The values of Co, Cl,


AO=
and


a ai = 0.121, Al = 0.2
C2 may be determined as


and A2 = -0.1.


Co = 0.212

Cl = 0.181

C2 = -0.1


and from equation (29), a =
If the resulting value of a
affect the values of the B


0.567, which checks with the assumed value.
differs enough from the assumed value to
coefficients, step 2 should be repeated.


Step 5


From equation (60) the value of c is determined as


E = 0.0015 radian = 0.0650


Step 4


From equation (5l),


CLl = 0.251


CLc = 0.072


Therefore,


CD = 0.251 tan(12.0650) + 0.072 tan(15.920) + 0.004 = 0.075


Cf = 0.004






NACA RM L57116


CENTER OF PRESSURE


Step 1

From step 2 of the drag-coefficient calculation,


CO = 0.212 Ci = 0.181 C2 = -0.1


Thus, from equation (62),


xc.p.,l = 0.4c


Step 2

From equation (64) and by using


CL, = 0.251


and


Lc = 0.072


it is found that


xc.p. = 0.45c





NACA RM L57Il6


REFERENCES


1. Tulin, M. P., and Burkart, M. P.: Linearized Theory for Flows About
Lifting Foils at Zero Cavitation Number. Rep. C-658, David W. Taylor
Model Basin, Navy Dept., Feb. 1955.

2. Green, A. E.: Note on the Gliding of a Plate on the Surface of a
Stream. Proc. Cambridge Phil. Soc., vol. XXXII, pt. 2, May 1956,
pp. 248-252.

5. Perry, Byrne: Experiments on Struts Piercing the Water Surface.
Rep. No. E-55.1 (Contract N125s-91875), C.I.T., Hydrod. Lab.,
Dec. 1954. (Available from ASTIA as AD No. 56179.)

4. Eisenberg, Phillip: On the Mechanism and Prevention of Cavitation.
Rep. 712, David W. Taylor Model Basin, Navy Dept., July 1950.

5. Wadlin, Kenneth L., Ramsen, John A., and Vaughan, Victor L., Jr.:
The Hydrodynamic Characteristics of Modified Rectangular Flat Plates
Having Aspect Ratios of 1.00, 0.25, and 0.125 and Operating Near a
Free Water Surface. NACA Rep. 1246, 1955. (Supersedes NACA
TN's 5079 by Wadlin, Ramsen, and Vaughan and 5249 by Ramsen and
Vaughan.)

6. Lamb, Horace: Hydrodynamics. Reprint of sixth ed. (first American
ed.) Dover Publications, 1945.

7. Milne-Thompson, L. M.: Theoretical Hydrodynamics. Second ed.
MacMillan and Co., Ltd., 1949.

8. Rosenhead, L.: Resistance to a Barrier in the Shape of an Arc of
Circle. Proc. Roy. Soc. (London), ser. A, vol. 117, no. 777,
Jan. 2, 1928, pp. 417-455.

9. Wu, T. Yao-tsu: A Free Streamline Theory for Two-Dimensional Fully
Cavitated Hydrofoils. Rep. No. 21-17 (Contract N6onr-24420), C.I.T.,
Hydrod. Lab., July 1955.

10. Wagner, Herbert: Planing of Watercraft. NACA TM 1159, 1948.

11. Weighardt, Karl: Chordwise Load Distribution of a Simple Rectangular
Wing. NACA TM 965, 1940.

12. Glauiert, H.: The Elements of Airfoil and Airscrew Theory. Second
ed., Cambridge Univ. Press, 1947. (Reprinted 1948.)






NACA RM L57116


15. Johnson, Virgil E., Jr.: Theoretical Determination of Low-Drag
Supercavitating Hydrofoils and Their Two-Dimensional Characteristics
at Zero Cavitation Number. MACA RM L57GIla, 1957.

14. Jones, Robert T.: Correction of Lifting-Line Theory for the Effect
of the Chord. NACA TN 817, 1941.

15. Shuford, Charles L., Jr.: A Theoretical and Experimental Study of
Planing Surfaces Including Effects of Cross Section and Plan Form.
NACA TN 5959, 1957.

16. Flax, A. H., and Lawrence, H. R.: The Aerodynamics of Low-Aspect-
Ratio Wings and Wing-Body Combinations. Rep. No. CAL-57, Cornell
Aero. Lab., Inc., Sept. 1951.

17. Fuller, Roger D.: Model Experiments With Hydrofoils and Wedges for
Rough Water Seaplane Design. Rep. No. ZH-102 (Contract
NOa(s)-12145), CONVAIR, July 1955.

18. Parkin, Blaine R.: Experiments on Circular Arc and Flat Plate
Hydrofoils in Noncavitating and Full Cavity Flow. Rep. No. 47-6
(Contract Nonv-220(12)), C.I.T., Hydrod. Lab., Feb. 1956.

19. Newman, J. N.: Super Cavitating Flow Past Bodies With Finite Leading
Edge Thickness. Rep. 1081, David Taylor Model Basin, Navy Dept.,
Sept. 1956.






NACA RM L57116


a = Po c Pv for vapor cavitation
(1 q


Foil(a) Supercaviating flow at great depth of submersion.PC
cavi ty




(a) Supercavitating flaw at great depth of submersion.


Water surface


o 0


(b) Supercavitating or ventilated flow near the free surface.

Figure 1.- Definition sketch.







NACA RM L57I16


I TT I












& 4


tof
I I*T

00










1* I4 I F1 .. .a L


CM BO 3 0 '0 (01


1.BUt~up9J cefIj


d)

43



0
Pi









43
A
(D








43 r
S 0










-4.r
rO


o m








ii
COO(



4 '3
a 4-1 r


o "e
0



a c-



Ot
43
0 dl
a E








III
S 04

6 l0








ir-4



43
o a)












M -H
t oin


041

CO *



I ,C


*-pi

*P
{ 1


0






NACA RM L57116 55






0
.r4


r-1
0
I"I


+3
I I a
















\ u! O/
C .. 0

0
















I I I
to






































9ep 0 '&ire jRe0dB
0





)0



0



to t-o
Ln0






ga 0a~ ed







56 NACA RM L57116










.ao

0

*0 0




I d u
," / (fl

I 4 0

0.c 0

0)0


10

0.


/ H
/ o cn 00*
1-1~. yh








3 0
I I


f *>(" C.) r4













0
,/./~4 -H'u u'







4-O
o 0\0








0 r0
q-*k





a.
tio
\o





Qc)



C-i
// I I "









i.. .- 1 l 1
a. (*f





NACA RM L57116


Water surface


Upper cavity surface
,-u = 0
u=0

\Lower cavity surface=0
Lower cavity surface


V = V


(a) Hydrofoil, Z-plane.


Transformed upper
cavity surface
U =0 -


U =0


Transformed ,
water surface
xy = d/2


Center of pressure


Transformed lower
cavity surface


(b) Equivalent airfoil, Z-plane.

Figure 5.- Linearized boundary conditions in hydrofoil and equivalent
airfoil planes.






NACA RM L57116


Transformed
water surface


Control point


Equivalent airfoil,


S- plane


Figure 6.- Linearized model for calculating effect of depth of submer-
sion on lift coefficient of flat plate (a = 0).




--Exact solution, Green (ot 0) V
0 Calculated from equation (23)


CL O O
CL, co------- 0 --- 0
-1.0 -
Ctd1


0 1.0 2.0


3.0 h.o 5.0


-Ratio of depth of submersion to chord, d/c


Figure 7.- Comparison of'linearized solution with exact
effect of depth of -submersion on lift coefficient of
(a = o).


soultion for
a flat plate





NACA RM L57116


4)
54
cl
0
in a





8^

I,


'S (
C
C


T'


+
IU
cd


1-



-H


LLt -U -
IC)|J


)
)
)
)
)













I-






NACA RM L57116


.3
----------III IIII IIII II- T -- -




de = 0


B0 2 -

,.071 5
1.0
1 .25-
-=:Z. 1 1 ::?:. L A 1.0 1- : >; : --- ---------I
z = -. _- 2 = : ; -



0
::: .0:': 9.0- !~ "r^ !!! =! ====! "
.1 .2 .3 .5 .6
a, chords


(a) BO.

LL




.3


-d/c 0 -- --

Bl 2 -









0
::^ ^2 -y --- -^= ^- .^;-------------- 1 : = ._ _






.1 .2 .3 .. .5 .6
a, chords


(b) Bl.

Figure 9.- The B coefficients.






NACA EM L57'I16


a, chords


(c) B2.


.h
a, chords

(d) By5


a, chords


(e) B4.

Figure 9.- Concluded.


0






1.0




B3 .5


o u
.15










0 -







62 IACA M L57116







0




C,

0%
II


Sa,
aa


CD

a O
1*.
1. 1 1 1 10
11 f i ll I I


4- 4.





0 Cl
^ ^ ^ ^ - --^^ ^ V A d ):: : :*











o 1-io
-- -- -9-- - - -I














to



1-1
I w m
- - --. -__ --__ k -

-- -- -. -- ----- H o 0^ rd- o o



























fa I
---------- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ .- A-- -- --. -- ._ -. T I -












--------- --- 0 -









93

'-4
.- ,-4 1 .- |







Al ~ rz4







NACA RM L57116


'-n




0

0
.4
43



.4

II


~4-

I





4c
ml
0




a
U
U

0
I.
a
9-I
a
U
I.
.94
U


... ... ... ..


9-

-0












-t
a


.4
5-:: :


11 1 1 F1,4 ~I I I I I I &.


. I . .- ,


0
1-4


- -






-------------
T I
4

mill-
------- -----------


---lL LLLL. I


018
6s


U
d


0)

a
Ch flp








o. E





bc1
0
4-)







4-)














D



43
CIS

0
a)

















00 r- .
-4
oi
*0 CIS
u
+0
0


0
0
*o



U)








0)



CH-
o 0

9
p. Q












0


r4
0 CH
SIn


0




p4,

0
0L)


a ,

C.-

I-

9-I
0o-


t I I I I


------- ----



-TT T T F


I I I I I I I i I I






NACA RM L57I116


Water surface




Stagnation streamline


Stap-nation streamline


8- C =


(a) 5/c = -.


Free surface I T


(b) 5/c = 0.


(c) O < 5/c < oo.


Figure 12.- Location and direction of stagnation
normal to direction of motion.


line on flat plate






NACA RM L57116


Water surface


(a) Hydrofoil, Z-plane.


Transformed upper
cavity surface


Transformed water
surface


*Upper cavity surface


(b) Airfoil,


Z = plane.


Figure 15.- Hydrofoil and equivalent airfoil at large angles of attack
and small depths of submersion.






NACA EM L5TI16


; M







stl
S..




14**


EPRR



r4 .

*R *



RP
.4


9-I


0 *

.9-"
0 *


0 *


II




II
GRE





.N
we.






in
O*00
%.oo
*- *

*..







Bn







8 M



.a
^0 8 *



1 0 *


pam .

rhs


02
>, <

oo


04

to




1410

I-


'a
*j .,; l


0ooC

WKh


I






NACA RM L57'116 67



















CC
a ,




V -

0 l




I a a
I' |Q6 o



rI
SO



,-4
4J 0)
'd ud



aa a i





NACA IM L57I16


Witndscreen


A- Ar bjOCLI I* i ^'
pop
4 ^ g


'.4""


L-57-2745
Figure 15.- Test set-up showing cambered hydrofoil with aspect ratio
of 1 and balance attached to towing carriage.


S-'


4.



V/k


ho







NACA RM L57116


Lift


- O -- Direction
i


Figure 16.- System of axes.

Cable to carriage


Pressure transducer -














Foil strut


Gauge strut

- Foil strut, L. E.


o.d.


(a) Side elevation.


Gauge strut


-li1
k-i-


Foil strut


(b) Front elevation.


Figure 17.- Schematic drawing of spray-thickness gauge.


Moment


of mot on


---=






NACA RM L57116


V = 20 fps V = 40 fps


V = 60 fps V = 80 fps


(a) a = 8.


V = 10 fps V = 20 fps


V = 50 fps V = 40 fps


(b) a = 100.


L-57-2740


Figure 18.- Flow about the


flat lifting surface, depth of submersion is
0.5 inch.






NACA RM L57TI16


V = 10 fps V = 20 fps


V = 50 fps V = 40 fps

(c) a. = 80. Air supplied at 0.012 pound per second for V = 10, 20,
30 feet per second.

Figure 18.- Concluded.









V = 20 fps V = 40 fps









V =60 fps V =80 fps

(a) a = 100. L-57-2741

Figure 19.- Flow about cambered lifting surface, depth of submersion is
0.5 inch.






NACA RM L57116


V = 20 fps V = 40 fps


V = 59 fps V = 80 fps


(b) a = 120.


V = 10 fps V = 20 fps


V = 50 fps V = 40 fps


(c) a, = 160.


L-57-2742


Figure 19.- Concluded.





NACA RM L57I16


20
t0
S



a 15
0


0 10
'-4


Canbered


5 h


Speed, fps


Figure 20.- Vortex ventilation speed for 0.5 inch depth of submersion.







NACA EM L57116


I0

j


(U
P4



+3

-~ H
S d

a5


* CS tq
0M a f *c


*
Cu
0
11
rc-r


0



rR
*H




-aa
5I"
0


,


d o


CN H

No


**^

*4

OW
o +?



OC4
01-

t1-4


w









ar..
kIL











00
0 *
CO
Pi(U

P*4













>71i
a4 *r
O -I



IOd
HO

43W
0 (U









"g
Ho3



,-1
4343P



* r4

0 i-i

S db



w
1-1





NACA RM L57116


__ __ [_________ I __ __


Depth,
in.
0 1/2
o 1/2
0 1/2
A 1/2
s 1/2
b 1/2
0 1/2
01/2
c0
- 1/2


I I
a,
deg Ventilation
20 Vortex
18 Vortex
16 Vortex
14 Vortex
12 Vortex
10 Vortex
8 Forced air
6 Forced air
4 Vortex
10 Forced air


//


a deg
120


1/


18


400
2
3 0----0/-
mm /' /// z^H^


/


I/


250 ------------ T /------


200--//


an_ ^-"^__
15





II


0 10 2 30 L0 50 60 70 8D 90
Speed, fps


(a) Lift.


Figure 22.- Characteristics of ventilated flat lifting surface.


-I-


550


500-


450 -


*0
.3
a-


30*


3w






NACA RM L57I16


Depth, V,
In. desg Ventilation
0 1/2 2D Vortex
o 1/2 18 Vortex
0 1/2 16 Vortex
S 1/2 14 Vortex -
1 1/2 12 Vortex
L 1/2 10 Vortex
o 1/2 8 Forced air
o 1/2 6 Forced air
0 0 4 Vortex
CS 1/2 10 Forced air


0 10 20 3D 40 50
Speed, fps


(b) Drag.


Figure 22.- Continued


_ 18


180

180 ---


100



80



60



40



20


90


16












14











6
------------------------ / / 4






NACA RM L57I16


Speed, fps


(c) Pitching moment about leading edge.


Figure 22.- Concluded.






NACA RM L57I16


Depth,
in.
0 1/2
o 1/2
0 1/2
A 1/2
L 1/2
1 1/2
Ci 1/2
or 0
Cy 0
cr 0


Ventilation
Vortex
Vortex
Vortex
Vortex ______
Vortex
Vortex, wire on L.E.
Forced air, wire on L.E.
Vorte. _____
Vortex
Vortex


/
Ii


q_ deg
A)




/14
1 2
/
/ 2


7/A


/ /
----------/ //211-


70 s0 90


40
Speed, fps


(a) Lift.


Figure 25.- Characteristics of ventilated cambered lifting surface.


1,000


900


800



700


6001


.0
1-4
.500

4W
40






NACA RM L57116


79







Ventilation
Vortex _______
Vortex
Vortex
Vortex
Vortex
Vortex, wire on L.E.
Forced air, wire on L.E. a, deg
Vortex
Vortex
Vortex

18



C16







12


10
44-6














40 50 60 70 80 90
Speed, fps


(b) Drag.


Figure 25.- Continued.







80 NACA FM L57I16








-360-

0 1/2 20 Vortex
D 1/2 18 Vortex
-320 -- 1/2 16 Vortex -
A 1/2 1u Vortex
N 1/2 12 Vortex
D 1/2 10 Vortex, wire on L.E.
-280 0- 1/2 8 Foreed air, wire on L.E.
C' 0 10 Vortex
do 8 Vortex a dg
20
V 0 6 Vortex
-240

16

-200 ---




-~ 112 1
-160



-120



-80



-40




0 10 20 30 40 50 60 70 80 90
Speed, fps


(c) Pitching moment about leading edge.


Figure 25.- Concluded.






NACA RM L57116


* doeg
0 20
0 16


0 2


) L0
Speed, fpe


) 8


Speed, fpa Speed, fps


(a) Lift.


Figure 24.- Characteristics of ventilated flat plate at
for depths of submersion of 0, 1.0, 1.5, and 2.0


a. = 160 and 200
inches.


Depth. d 0


-s

I
7


-240
-I


00 -


Speed, fps


inch


1







MACA RM L57116


d 1.5 Inches

120



80 ----



LIC




0 2D ) 6L I
Speed, fps


a. dog
E 20
0 16


160I

d 1.0 inches
120





40-^


0 0 0O
Speed, fps


60 s0


Speed, fpa


(b) Drag.

Figure 24.- Continued.






NACA RM L57116


t7,deg
0 20
016


-1Uj ---- ---- ---- ----


Depth, d 0 Inch
-80



-60



-'40
..}/


-20




S0 20 L40 60 8
Speed, rps






EJ -100
A.
d 1.5 inches

-s0



-60 --



-.LI ---



-20--
0


-100



-80



-60



-n0



-20




0 0







-100



-80



-60



-u0


O40 60 80 0
Speed, Dos


d 1.0 inch



















20 Lo 60
Speed. fos


wo
Speed. fos


(c) Pitching moment about leading edge.


Figure 24.- Concluded.






NACA RM L57I16


U
0
R

o'


9





.o
a

0

c, S
*




\ 8


\ ^


8

0
0



4a
\ AS5
S 0
\ I..


4)



V
ta
(U


p<
a!
A::





..)
4-4













to
*H
U3








43
O :





"* 4.
*-HO







4 0
p4








0 Ur

34k a
*s ,
0







%i
o o
r4 4 0
0* *H I.-







4-
4)











.1I


O 0
a)
%-4 i

0o b

'i ^


8 '8
0)


CM



o a,

--4
rz


SI

0


0 Poqo o su3oq AeDds jo o
.- *







NACA RM L57116


/



/*


/


gap 'o 'TUZuozTJoq Bqf qtlTA ReJds jo aejuy


,C 0
*
0




0
54



0


o

A








P g
* 4I
0



0


o50
U,
0:


0 0
R L


0
r-1


P4
+3
C)
p4






w
CO






ci
'C
4,








m
r-



o3
o
H
















r-4
C,















0


en
2'T















CO
.11
W





C


I
P43




4M
CH
Hq-,





0
rS


,0





-P
0)
43



CM

a)
.r-4








NACA RM L57I16


U') J.


iN r


'I 'Dlu9OTJJSoo IJn


0
r-4

4g
e 0


63 *

M-0
a-s



-24 04
X.1. 1-4


g00




S I






CNN
10

--4
01 .J-
c~iC-


to
.4.4









a)
0
*pO

P 0



*J2
4i-







aW,




C4


r- (a
*H 00
04 w
4 *H





OH
-4 4C 3




ot (U
ri-4



*H















00
4 .


cud




r C*H
a) 0)








CI)


0 Q)
*4
0, 'd
.P
cd*








*rii
4.) (U)






P44
toi!











[0
IU.

I-pc




*In






fr-i






NACA RM L57116


0 0
do 0 0
CM r-t


CM


70'4ueloTJJlao 4JTI


H 0

43

0

Wa
\r' *B (U
CMO
43P


ad

0
cu
HH

0
V 0 C)
'd u ..>



-4 HOI

s^
0
,o


-r- i -1 1
M ^
0
*H H II



a g0


o0 al





a ca
a) a)r





0 0
~0 He
0~


o 0


P




C)
o b

4,
x o
o


|S

o c

(U

0b

Z11


4P


(
El

0







88 NACA RM L57I16



SOOO A Experiment
Theory

t.4




00
Op
0

U
.2-



.1 -




0 .4 .8 1.2 1.6 2.0 2.4 2.8 3.2
Ratio of depth of submersion to chord, d/c


(a) Cambered section, a = 90.


.30



.25 -'c


CC
15


.20 12o



U1 190

10 O go
%Z 60
10CA 0



.05 -




0 .4 .8 1.2 1.6 2.0 2.4 2.B 3.2
Ratio of depth of submersion to chord, d/c


(b) Flat plate.

Figure 29.- Comparison of present theory with experimental data of
Fuller obtained on hydrofoils having an aspect ratio of 2.







NACA RM L57I116


4J *J +

4 4) 04
ewe,



ew


.CL '
Si r-- e-4

0Cl cr-
0t-


N 0

% 'uBio1jjooo fuia


R
4-)



i
;4





O
co-


(U
ri-
CJ

0J
*4
4-i







rd U
r-I w




to H S0




4- P

0 m a] tio
rdu











.4-,
l Ud









lO 0d
4--)pr
CU














0 0
gjr-4











*d
0 C













O OCU
U)




















0
'-


oo
1-4




















S.r

aP ,
00 U *
*r I (
.P'd-l




^5-5



0 U



*r4

5-
aS
*H
I
0


(U

0

taO
*H-
F'







NACA RM L57116


18





16





12





10



0"
-4



I.,
4-I




t





4


(ref. L)


.3 AL
Lilt coejlicient, CL


Figure 51.- Comparison of theoretical and experimental ventilated lift-
drag ratios for flat and cambered lifting surfaces with aspect ratio
of 1.


Depth,
in.
0 0 0 1/2 Experiment, vortex ventilation
0- 1/2 Experiment, forced air ventilation
S 0 Experiment, vortex ventilation

Note: Experiment and theory include skin friction,
(C = 0.00oo4)




2









h
.' I












6

6
In. -Camfrred section
8








10

Theory, 10 12
^ LL







I in 1ii
S' 16
% 18
16 -. 20


010







NACA RM L57I16


S
I..


o k.


am


93
0










a 0
a~i id
-I







4- N
-.w4

o o-


0



D-


ir\ a vr N -
* *
pjoqo jo WSozeod
jo ej6ugo 0% S9ps SaIp9S[ UUJJ SOoB=%'


03
4)'
Ea
+3

r*






rd
C)










2 so
MM












oO
aO m




*r-












W
0
*-

*








-4 C























V.
4.1


00
o ^ c)
-' C d
r)




0 <

4 3

00^




S0



C)


(I


'anssea.zd







NACA RM L57I16



Air %eed. Vap
Cavity Mirty

So 2D
0A 80r

60


(a) a = 160.


upercavitration -Cabered

Flat


-- Partial cavitation
OUL (Both models) a 4
I IlI ili


0 .2 .4 .6 .8


a. = 18 0.


1.0 1.2 1.4






SCanbered

A -Flat


Cavitation number, o PC


(c) a = 200.

Figure 35.- Effect of cavitation number on lift coefficients of flat
and cambered lifting surfaces. Depth of submersion is 6 inches;
aspect ratio of 1.


Hat. 00miboll for
are -flao aed






NACA RM L57I16


Air, a = 0.59, V =20 fps


Vapor, a = 0.3355, V = 80 fps


(a) Flat plate.


L-57-2745


Figure 34.- Comparison of air- and vapor-filled cavities for depth of
submersion of 6 inches and a, of 200.







NACA RM L57I16


Air, a = O.48, V = 20 fps


Vapor, a = 0.55, V = 80 fps

(b) Cambered section.

Figure 34.- Concluded.


L-57-2744


NACA Lagley Frlad, Va.
















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