Theoretical determination of low-drag supercavitating hydrofoils and their two-dimensional characteristics at zero cavit...

MISSING IMAGE

Material Information

Title:
Theoretical determination of low-drag supercavitating hydrofoils and their two-dimensional characteristics at zero cavitation number
Series Title:
NACA RM
Physical Description:
29 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: Two new low-drag hydrofoils for operation at zero cavitation number are derived by the application of the Tulin-Burkart linearized theory. The two-dimensional lift-drag ratios of these two hydrofoils operating at design angle of attack are shown to be, theoretically, about 45 and 80 percent greater than the Tulin-Burkart configuration. A simplified calculation of the location of the upper cavity streamline for arbitrary configurations is also presented.
Bibliography:
Includes bibliographic references (p. 23).
Additional Physical Form:
Also available in electronic format.
Statement of Responsibility:
by Virgil E. Johnson, Jr.
General Note:
"Report date September 30, 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 - 003834476
oclc - 150579969
System ID:
AA00009188:00001


This item is only available as the following downloads:


Full Text



CL





PX'

4A o





'5 itq I






"V


I e It$




Pit A

44



4 L





-you All
0, v Fr




lt,"t ',t I w




IVA!


0 twtl4A, y*t'


"Saw.
I





It,





t 1 4











OT l

Tt

"''t
T`nlr*

Mw


t,"r'
16"', mttt
out S' Of me,











40
d Val






NACA RM L57Glla

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


RESEARCH MEMORANDUM


THEORETICAL DETERMINATION OF LOW-DRAG SUPERCAVITATING

HYDROFOILS AND THEIR TWO-DIMENSIONAL CHARACTERISTICS

AT ZERO CAVITATION NUMBER

By Virgil E. Johnson, Jr.


SUMMARY


The linearized theory of Tulin and Burkart for two-dimensional super-
cavitating hydrofoils operating at zero cavitation number is applied to
the derivation of two new low-drag configurations. These sections were
derived by assuming additional terms in the vorticity distribution of the
equivalent airfoil; in particular, three and five terms were considered.
The characteristics of both the three- and five-term airfoils are shown
to be superior to the Tulin-Burkart configuration. For example, the two-
dimensional lift-drag ratios of these new sections operating at their
design lift coefficient are theoretically about 45 percent and 80 percent
greater than the Tulin-Burkart configuration.

A simplified calculation of the location of the cavity boundary
streamline for arbitrary configurations is also presented. The method
assumes that the contribution of camber to the equivalent airfoil vor-
ticity distribution is concentrated at the center of pressure.


INTRODUCTION


In reference 1 Tulin and Burkart present a linearized theory for
determining the characteristics of supercavitating two-dimensional hydro-
foils of arbitrary section operating at zero cavitation number. It is
shown that the hydrofoil problem may be transformed into an equivalent
airfoil problem which can be treated by well-known thin-airfoil theory.
The theory shows that hydrofoils with high lift-drag ratios are those
whose equivalent airfoils have their centers of pressure as far aft as
possible while maintaining all positive vorticity over the chord. In
reference 1 such a low-drag section was chosen by specifying only two
sine terms in the pressure-distribution expansion (or equivalently, the
vorticity distribution) and then these two coefficients were adjusted
so that the necessary conditions for high lift-drag ratios were satisfied.






NACA RM L57Glla


The purpose of the present report is to derive hydrofoils whose air-
foil center of pressure is further aft than the Tulin-Burkart configura-
tion and thus even higher lift-drag ratios are obtained. One obvious
means of accomplishing this is to specify a given pressure distribution
on the airfoil and then to determine the Fourier coefficients which
describe it. This procedure usually will not lead to a closed-form
expression for the airfoil or hydrofoil shape; however, the method does
permit adequate solutions in tabular form to be made. In the present
case, it was reasoned that superior configurations could be derived
merely by choosing more terms in the vorticity series expansion and
adjusting the coefficients for maximum lift-drag ratio exactly as was
done by Tulin and Burkart. In this manner, the results would be in a
closed form. The number of terms chosen for the analysis was specified
as three for the first case and five for the second case. The results
of calculations based on these presumptions are presented.

Since, for practical reasons the hydrofoils must have some thick-
ness; the shape of the cavity streamline leaving the leading edge is
required. The thickness of the hydrofoil that can be permitted is then
such that the hydrofoil upper surface lies below this free streamline.
The linearized theory permits, in principle, this streamline location
to be calculated; however, when the expression for the airfoil vorticity
distribution becomes very lengthy, the calculation is very difficult.
If the vorticity due to camber is assumed to be concentrated at the cen-
ter of pressure of the airfoil, the problem is greatly simplified. The
results of an analysis based on this assumption are also presented.


SYMBOLS


An coefficients of sine-series expansion of airfoil vorticity
distribution; that is,

0(x) = 2V (Ao cot + Al sin B + A2 sin 29 An sin n)
2


A' value of A0 when a.= 0; that is, A0' f a- d9
0A 0 dR


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


a2 A= 2

A5

az =
3 Al






NACA RM L57Glla


c chord

CD drag coefficient, D/qc

CL lift coefficient, L/qc


CL,d design lift coefficient at a = 0

Cm pitching-moment coefficient about leading edge, M/qc2

C0m third-moment coefficient about leading edge, M /qc4

p p
Cp pressure coefficient, ---qP

D drag force

k
k number of terms in summation An sin n9
nM
n=l

L lift force

M pitching moment about leading edge


M3 third moment about leading edge, 2 p(xi)idi
-0

p local pressure

p ambient or free-stream pressure


q dynamic pressure, pV2


I distance from section reference line to upper cavity streamline

V speed of advance, fps

u perturbation velocity in x-direction

v perturbation velocity in y-direction






NACA RM L57G11a


x' dimensionless distance parameter along X-axis, x/c

x distance along X-axis

y' dimensionless distance parameter along Y-axis, y/c

y distance along Y-axis

a geometric angle of attack, radians

P circulation

Q vorticity

6 variable related to distance along equivalent airfoil chord
by equation x = 2 c(l cos 9)

Subscripts:

a due to A0 or if Ao' = 0 is due to angle of attack

c due to camber

Barred symbols refer to quantities in the airfoil plane and unbarred
symbols to quantities in the hydrofoil plane.


SUMMARY OF THE TULIN-BURKART LINEARIZED THEORY


Since it will be necessary in the derivation of the new hydrofoils
to refer frequently to the linear theory of reference 1, a summary of
the principal results of that theory is useful.

In reference 1 it is shown that the problem of a hydrofoil operating
at zero cavitation number in the Z-plane may be transformed into an air-
foil problem in the Z-plane by the relationship Z = ZF7. By denoting
properties of the equivalent airfoil with barred symbols and those of the
hydrofoil with unbarred symbols, the following relationships are derived:

d&R)= 2(R2) (1)
di dx

() = u (2) (2)






NACA RM L57Gl1la


CL = m = A0 + Al T


CD = L2)


= A A 2


Cm = C,5 = 32(o + 71 7A2 + )A z)


The coefficients An are the thin-airfoil coefficients in the sine-
series expansion of the velocity perturbations U(i) where


u() = V Ao cot 2 + A sin
+ L
\ n=l


where


I = 1 (1 cos 9)
2


(o 9 6 x)


Since U(R) = 2n(R), equation (6) defines the vorticity
on the equivalent airfoil as


I w,
() = 2V Ao cot +
n=l


distribution


An sin no)


The values of the A coefficients can be found for a
from the following equations:


given configuration


Ao = 1
.n f o


SdO + a = a + Ao'
dx


An = I d cos nO dO
IC f dA


(9a)


(9b)






NACA RM L57Glla


The first term in equation (8); that is, the Ao term, is the
vorticity due to angle of attack and the second term is that due to
camber. In order to isolate the effects of camber, Ao will be con-
sidered zero. Any section profile derived on this basis will also, for
convenience, be oriented with respect to the X-axis in such a manner
that Ao' = 0. From equation (9a) these conditions require that a
also be equal to zero. Thus, the derived orientation is defined as the
zero-angle-of-attack case.

When A0 is set equal to zero, the hydrofoil lift-drag ratio for
a given lift coefficient is obtained from equations (5) and (4) as
follows:
2

CL 2(A-%) = A2 (10)
CD A/4 2CL 2 Al)


A2
Obviously, for maximum lift-drag ratio, must be as large as possi-
A1
ble. However, if the assumed condition that a cavity exists only on the
upper surface is to be real, the vorticity distribution given by equa-
tion (8) must be positive in the interval 0 S 9 5 v; that is, the pres-
sure on the hydrofoil lover surface must be positive over the entire
chord, otherwise a cavity will exist on the lower surface. Thus, for

maximum hydrofoil lift-drag ratio, A2 must be as large as possible
Al
and still satisfy the condition that

00
() = 2V An sin n G 0 (o0 9 5 ) (11)

n=l

With the stipulation that the vorticity distribution is defined by
only two terms in equation (ll), reference 1 finds the optimum relation-

ship between Al and A2 as = 1. This results in a hydrofoil
configuration given by the equation


=- + 5/2 ()1 (12)






NACA RM L57G11a


From equation (5) the design lift coefficient (that is, for a. = 0) for
this section is


CLd 5tA (15)


and the lift-drag ratio for this condition as obtained from equation (10)
is

L _25/ I (14)
D 7 2CL


Since 7/2CL represents the lift-drag ratio of a flat plate, the config-
uration given by equation (12) has a lift-drag ratio 25/h times as great
as that of the flat plate. When the hydrofoil given in equation (12) is
operated at an angle of attack, the lift-drag ratio becomes

a + 2 CLd
L I 2 (15)

( + CL,d)
The present analysis is concerned with the derivation of two new
configurations. The problem is exactly the same as that discussed in
reference 1 and summarized above except that the vorticity distribution
given by equation (11) is defined by: (1) three terms and (2) five
terms.


DERIVATION OF LOW-DRAG HYDROFOILS AND THEIR CHARACTERISTICS


Statement of Problem

The problem under consideration is (1) to find the values of the
coefficients in the vorticity equation

k
n() = 2V A sin n 0 (o0 9 it) (16)

n=l


such that A2 is a maximum for the specific cases of k = 5 and
Al
k = 5, and (2) to use the method of reference 1 to find the shape of






8 NACA RM L57Glla


the hydrofoil which when transformed to the airfoil plane has the vor-
ticity distribution given in equation (16).


Three-Term Solution (k = 5)

For the case k = 5, the vorticity distribution given in equa-
tion (16) becomes


Q(R) = 2V(Al sin 9 + A2 sin 29 + A3 sin 30) 0 (17)


The solution of equation (17) is obtained in the following manner. Let


a2 = (18)
Al


a = A (19)
SAl

The problem is now to find a2 and a so that a2 is a maximum and


sin a2 sin 29 + a sin 30 5 0 o 9 ) (20)


Substituting trigonometric identities for the functions of the multi-
ples of 9, equation (20) may be written as


1 2a2 cos + 5a ha5 sin29 0 (21)


The minimum of equation (21) occurs when

-1 a
9 = cos 2
4a5


Substituting this value of 9 into equation (21) gives

2a22 / 2 1
+ 5a a al- 2 0 (22)
a5 5 \ l+a 3






INACA RM L57Glla


or


ha- 4aa2 a22 >0 = b (b 0) (25)


Therefore,

a2 =t +4a ha2 b (24)


and the term under the radical has a maximum at a = Thus,
2

a- = t 4l --b (25)


and the maximum possible value of a2 is 1 which occurs when b = 0
1
and a = -. Since these values are obtained by considering the mini-
mum value of the vorticity or pressure on the airfoil, the condition
(x) 0 is satisfied for all values of 6 (0 5 6 it). Thus the
solution for the vorticity distribution for the case k = 5 is


Q(5) = 2VAl sin 9 sin 29 + 6 sin 59) (26)


The airfoil slope which has the vorticity distribution given by
equation (26) is obtained from reference 2 and is given as follows:


d- =Al cos 6 cos 29 + I cos 5I) (27)
dA (C 2 0)

Substituting trigonometric identities for the functions of the multiples
of 6, equation (27) becomes


dy- Af2 cose 2 cos2 1 cos 6 + 1 (28)



and since cos 6 = 1 2 .






NACA RM L57Glla


d ) = A, 21 2
dx


-2(l1 2 )


The slope of the equivalent hydrofoil is obtained from reference 1 and
is given as follows:


S(x) = -
dx d (


(30)


Equation (50) states that the slope of the hydrofoil can be obtained
from equation (29) by replacing i with 4x.

Thus, since E = Fc,


S= A 1[2 (1 2


- 2(1 2 2


Integrating from 0 to x and dividing both sides by c gives the desired
nondimensional hydrofoil shape; that is,


+ 8o(0)2 64()5/2


(52)


)= 51 20
c 10 (2/ ^c


By using equation (5), the lift coefficient of this hydrofoil becomes


CL + L


(35)


or for a = 0 the design lift coefficient is


CL,d = 4nA


(34)


The following drag coefficient may be obtained by using equation (4):


C = (.A)2


+2L 2
33ff /


(55)


(29)


- 1 2 +1


(51)


-c11-2 + 1






NACA RM L57011Ga


For a = 0, the lift-drag ratio is


L = 9( (56)



This value is nine times as large as that for a flat plate and 1.44 times
as large as the value for the hydrofoil of reference 1 where L = J .
D 4 k2CL)
The following lift-drag ratio may be obtained for finite angles of attack
(eqs. (52) and (34)):



L a + CLd7)

( + 2CLd)2



Five-Term Solution (k = 5)

For the case k = 5 the problem is to find the coefficients in the
following equation:


Q(f) = 2V(Al sin 9 + A2 sin 29 + A sin 59 + A4 sin 49 + A5 sin 50 (38)

A2
so that Q(0) o 0 and is a maximum.
Al

First attempts at a solution were made on a Fourier synthesizer.
The synthesizer is an electronic device which is capable of generating
80 harmonics of a Fourier series and recording the summation of these
components over any desired interval. The amplitude and phase angle of
each harmonic generator is controllable. By using only the first five
components and zero phase angle, it was discovered that a solution with
A2
-A2 roughly equal to 1.6 was apparently possible. Unfortunately the
Al
sensitivity of the equipment was not sufficient to assure positive values
of the summation of components near the leading edge. However, the syn-
thesizer result was encouraging, since it showed that apparently there
was a considerable advantage to using five terms, and revealed some of
the characteristics of the solution; for example, the algebraic sign and






NACA RM L57Glla


relative magnitude of each term. The most helpful method for obtaining
the best results was that used in obtaining the three-term solution.
This was to find first the minimums of equation (58) in terms of the
coefficients. The term was then assigned a value and the other
A1
coefficients were determined analytically so that three of these control
points (possible minimums) were zero and the values of the others were
A2
examined. By varying the value of and the choice of control
Al
points, a solution was obtained. The method is admittedly one of trial
and error and, since the process is somewhat lengthy, the details are
omitted. The best solution obtained was


(x) = 2VAl(sin 9 sin 26 + -sin 5 2 sin 6 + 1 sin 9 (59)



In the course of deriving the solution it was proven that the value
of must be less than V2. Since in the solution given by equa-
Al

tion (39) the term -A2 has a value of 4/5 (very close to the estab-
Al
lished maximum), further efforts to find a better solution were not con-
sidered worthwhile.

By following the method used for the three-term solution, the shape
of the hydrofoil corresponding to equation (59) is obtained as follows:


S= 210) 2,240(5/2 + 12,600(2 30912 2 +


55,80)5 15,560( )7/2 (40)


By using equation (5), the lift coefficient of this hydrofoil may
be given as


CL = i + (41)


or for a = 0 the design lift coefficient is

CL,d = 5itA (42)






NACA RM L57Glla


The following drag coefficient is obtained by using equation (14):


CD = + 1) = 2 4



and for a = 0 the lift-drag ratio is


L 100/c (4)
D 9 CL


This lift-drag ratio is about 11 times as large as the value for a flat
plate and nearly twice as efficient as the configuration of reference 1.

For finite angles of attack,

2
L a + 2 CL,d
D % '2
( + CLd



Comparison of the Low-Drag Configurations

Shape.- The shapes of the two-, three-, and five-term configurations
given by equations (12), (52), and (o40), respectively, are compared in
figure 1. It is apparent in figure 1 that the location of maximum camber

moves toward the trailing edge as 2 is increased. This movement
Al
corresponds to moving the center of pressure of the equivalent airfoil
toward the trailing edge. It is shown in reference 1 that the limiting
A2
value of is 2 and that for this value all the lift is concentrated
A1
at the trailing edge.

An important point to note in figure 1 is the appreciable deviation
of the three- and five-term hydrofoils from the X-axis. A similar devia-
tion from the X-axis exists in the airfoil plane whereit was originally
assumed that the vorticity was concentrated along the X-axis. Evidently
the assumption is not as good for the higher term hydrofoils as it is
for the two-term configuration, particularly for large magnitudes of
camber. As a result, the linearized theory may be less accurate in pre-
dicting the characteristics of the new hydrofoils.






NACA RM L57Gl1a


Pressure distribution.- From equations (2) and (6) and the linearized
Bernoulli equation, it can be shown that the pressure distribution over
the hydrofoil chord for Ao' = 0 is


P Pa2


k
cot + An sin
n=1


(46)


or separating the
Cp ,a and camber,


two components
Cp gives


Cp


into contributions of angle of attack


= 2 cot
2


(47)


k
Cp,c =2A n sin n6
n=l1


(48)


In equations (47) and (48) the location on the hydrofoil corresponding
to a given value of e can be found from the relationship


x= 1(1 cos 9)
E 1T


since = ( )2 The coefficient A1 defines a particular value of the

hydrofoil lift coefficient at a = 0; that is, the design lift coeffi-
cient CLd given in equations (15), (54), and (42). Therefore, with
the aid of these equations, equation (48) can also be written in terms
of CLd as


Cp,c
CL,d


Al
S2 A
CL, d


(49)


k A s i n n O
Ax






NACA RM L57Glla


Thus, the total pressure distribution on the hydrofoils can be obtained
from


C pp = a + 'P CLd (50)


Equations (47) and (49) are plotted in figure 2 for the three hydrofoils
under consideration. It is apparent in figure 2(a) that the location of
the maximum pressure moves aft as A2 is increased. It may also be
Al
seen that the adverse pressure gradient to the left of the pressure maxi-
A2
mum also increases as A increases. Thus the five-term hydrofoil is
A1
more susceptible to boundary-layer separation than the other two. If
such separation occurs, the pressure distribution shown will be con-
siderably altered. This of course also applies to the two- and three-
term solutions but to a lesser degree. Because the adverse gradient
increases so rapidly with increase in -2, it is believed that further
A2 Al
increases in -, attained by considering more terms in the vorticity
Al
expansion, will not be practical.

The small pressure "humps" near the leading edge of the three- and
five-term hydrofoils are peculiar to the solutions found but could be
eliminated by proper adjustment of the coefficients. However, the exist-
ence of these humps is probably not important in a practical configuration.

Lift-drag ratio.- The lift-drag ratio and lift coefficient given by
equations (15), (57), and (45) are plotted for the three low-drag hydro-
foils in figure 5. The relationship L= -- for a flat plate is also
7 2CL
included. The solid lines show the lift-drag ratios of the three low-
drag hydrofoils when operated at a = 0 but for various magnitudes of
camber; that is, CL,d. The broken curves are for the particular magni-
tude of camber for which CL,d = 0.2 and 0.4, but the angle of attack
is varied.

In figure 5 it may be noted that the lift-drag ratios of the three-
and five-term solutions when operating at their design lift coefficients
are considerably higher than the two-term solution of reference 1. It
is also evident in figure 5 that the relative magnitude of the lift-drag
ratios of the three sections decreases with increase in angle of attack.
However, figure 5 shows that the reduction in L/D with increasing angle
of attack is lessened by using higher values of CLd. Only the shaded






NACA RM L57Glla


portion of figure 5 is considered of practical value because the hydro-
foil must operate at finite angles of attack as will be pointed out in
the following section.


APPROXIMATE LOCATION OF THE CAVITY BOUNDARY STREAMLINE


The desirability of operating as near the design lift coefficient
as possible is obvious from figure 3. Therefore, since the hydrofoil
must have some thickness, the minimum angle at which a hydrofoil with
finite thickness canr operate with a cavity from the leading edge is
needed. The angle can be determined from the linearized theory of ref-
erence 1 by determining the location of the upper cavity boundary. The
minimum angle at which the upper cavity streamline clears the upper sur-
face of a hydrofoil of finite thickness is the angle desired. An approx-
imate solution for the location of the cavity streamline is derived in
the following analysis.

It is shown in reference 1 that the slope of the cavity upper sur-
face formed on a two-dimensional hydrofoil operating at zero cavitation
number and infinite depth can be obtained by transforming the vertical-
velocity perturbations ahead of the equivalent airfoil to the cavity
upper surface. These velocity perturbations are obtained by setting up
the expression for the velocity induced at a point -i, upstream of the
equivalent airfoil. The procedure usually leads to very complex prob-
lems in integration, particularly if the series expansion of the vor-
ticity distribution contributed by the camber is very lengthy. This
complication is avoided in the analysis to follow by assuming that the
vorticity contributed by the camber is concentrated at only one location,
the center of pressure of the airfoil when Ao = 0. The magnitude of
the concentrated circulation is similarly prescribed. The method can be
expected to give only an approximate answer, particularly if very much
of the camber vorticity is located near the leading edge. However, for
the new low-drag cambered sections being considered, the vorticity due
to camber is in fact concentrated away from the leading edge (as indi-
cated by fig. 2) and the approximation should be very good.

The hydrofoil and its equivalent transformed airfoil are shown in
figure 4. The symbols used in figure 4 are those used in reference 1
where u and v are the velocity perturbations in the x- and y-directions
in the hydrofoil plane, and a and V are the perturbations in the air-
foil plane induced by the airfoil circulation. In the airfoil plane the
vorticity is divided into two components


a = 2VAo cot (51)
0VA 2






NACA PJRM L57Gila


(52)


c = 2V ) An sin no
n=1


The first of these, :, is taken to be distributed over the chord
exactly as given by equation (51). However, to simplify the problem,
the second component of vorticity is assumed to be concentrated at one
point on the chord the center of pressure when Ao = 0. This point
is given in figure 4 as a distance aE aft of the leading edge. The
magnitude of the concentrated vorticity is denoted as c (circulation
due to camber) and is given by the following equation which is obtained
from thin-airfoil theory (for example, see ref. 2):


r. = TcV


(55)


The velocities induced by the circulation contributed by
as 7a, and those due to rc are denoted as vc.


A. are denoted


The slope of the cavity upper surface in the hydrofoil plane is


dyx v(x) V(-) V V(- + (5
x V V V V


and therefore


fx c x)
O V


Ix f a ( dx +)
Odx +
Jo v


(55)


The first term of equation (55) has been evaluated in reference 1;
therefore, only the contribution due to camber need be considered here.

In figure 4 the induced velocity at any point along the X-axis due
to the circulation rc is






MACA RM L57G11a


c11) = i7V A1
2jtF(a.)
2i5a


A1 V
a -
5


(56)


(57)


At a point forward of the airfoil -x,


e(-) = 'c-F ) = 1 Al V
4-
a + v

since E = Fc.

From equations (55) and (57)


A1
~4c


ca +


where the center of pressure of the airfoil
airfoil theory, for Ao = 0, as


- a loge ( )
+-.


(58)


a is found from the thin-


a = 1 -
2 (l-2 A,)


(59)


By combining equation (58) with the linearized flat-plate solution of
reference 1, the complete solution for the shape of the cavity upper
streamline on arbitrary configurations is


y' = Ao -x' + j 1 + 2F (x' + x') + I logl + 2rx'


A l a loe+ ax+
T[^-~a loge--


- 2 + ii' +


(60o)


where x' and y' denote the dimensionless parameters, x/c and y/c.
In this equation y is the distance from the X-axis to the cavity upper
surface. When the hydrofoil reference line is at an angle of attack,


Sc


= T






NACA RM L57Glla


the actual distance I from the reference line to the cavity streamline
is


I = y + ax


(61)


S= y + ax
c


as can be determined from figure 4. When equation (60) is
into equation (61) and Ao is replaced by its equivalent
equation (8), the following equation is obtained:


substituted
a + Ao' from


c= A'x + (a + A') + 2x x' + + + x -



2x'+ +] -a loge(a +aF)


(62)


where I/c is the dimensionless-distance parameter from the hydrofoil
reference line to the cavity upper surface.

By separating the angle of attack and camber contributions, equa-
tion (62) may be written as


= + (!c =


where for the case of
under consideration,


(65)


a Al


A0' = 0, which applies to the low-drag hydrofoils


l = (I + 2%) (x' + 1) + 1 log el + 2x' 2 x' + x
(ci 4 & x


and


c 1 r a + Ix'-\
-- F = S a loge
Al = a aa






20 NACA RM L57GUa


For each low-drag hydrofoil, A, may be replaced by its equivalent
in terms of CLd and equation (62) may be written as



S 4 -CLd(64)


where


-1= K S Ka loge a + x'
cL,d a


and K = Al which is for the Tulin-Burkart design, -2 for the
2CL,d 5r 53r
three-term design, and -- for the five-term design.
57

The value of a may be determined from equation (59) and is 5/8,
5/4, and 5/6 for the Tulin-Burkart, three-term, and five-term hydrofoils,

respectively. In figure 5(a), -a is plotted against x/c and in


figure 5(b), a is plotted against x/c for each of the low-drag
CL,d
hydrofoils. It is important to note the relative magnitudes of the

(i.) (GO
coefficients -E and a for a given value of x/c. At the trailing
a, CL,d

edge, is roughly 10 times as great as ---. This means that
a CL,d
except for small angles, the angle of attack is predominant in pre-
scribing the cavity shape.

The adequacy of the assumption of concentrated camber vorticity is
shown in figure 5(b) by comparing the solid (A) curve with the dashed
one. The solid curve was computed from equation (58) and the dashed
curve obtained from the coordinates given in reference 5. The tabulated
coordinates of reference 5 were computed for the Tulin-Burkart section by
considering the vorticity to be distributed as given in equation (52) and
performing the necessary complicated integration.






NACA RM L57Glla


In figure o the cavity shape derived from equation (64) for the low-
drag hydrofoils is shown for CLd = 0.2. Also shown in figure 6 is the
lower surface of each design for the value of CL,d = 0.2. An inter-
esting point (first noted in reference 5) is revealed in figure 6. The
calculated cavity shape at the design angle of attack falls beneath the
lower surface of the configuration. This result was not expected for
these low-drag hydrofoils because the camber was selected to have posi-
tive pressure everywhere on the lower surface. It is believed that the
disagreement is due to the inability of the linear theory to accurately
predict the pressure distribution when the airfoil vorticity is not in
reality distributed along the K-axis. However, the shape of the cavity
as determined from the linear theory is much less sensitive to the devia-
tion of the true location of the vorticity from the X-axis. That is, the
distance from a point on the equivalent airfoil to a point forward of the
leading edge is approximated very well by only the R component of the
distance. Thus, it is seen that the pressure distribution predicted from
the linear theory will be more nearly correct when the equivalent airfoil
is at an angle of attack and more symmetrically located about the X-axis.
It appears, then, that low-drag hydrofoils such as those derived in the
present paper and reference 1 can never be operated at the design angle
of attack for the following two reasons: (1) an upper surface cavity
will not form even on an infinitely thin configuration and (2) some thick-
ness must be provided for strength. The possibility that, near the design
angle of attack, the pressure distributions shown in figure 2 are incorrect
has been indicated by experimental investigation in reference 4. Even at
an angle of attack of 20, cavitation was found to occur near the leading
edge on the lower surface of the Tulin-Burkart configuration used in the
investigation.

Because of the need for operating at finite angles of attack, the
upper portion of figure 5 has been shaded to indicate that the lift-drag
ratios calculated near the design lift coefficient are of academic interest
only. In general, the minimum angle at which supercavitating flow from the
leading edge is possible will be equal to or greater than about 20. The
exact minimum angle and, thus, the practical range of operation will be
determined by the type and magnitude of camber and the thickness required
for strength.

In figure 6 the cavity streamline shown may be considered as possible
upper surfaces of practical hydrofoil configurations. For a given angle
of attack the five-term hydrofoil permits a thicker leading edge and a
more uniform section. These features are desirable structurally.








NACA RM L57G11a


CONCLUSIONS


The principal results obtained in the application of the linearized
theory to the design of new configurations may be summarized as follows:

1. The two-dimensional lift-drag ratios of the two new sections
operating at their design lift coefficient are theoretically about 45
and 80 percent greater than the Tulin-Burkart configuration.

2. The relative magnitude of the lift-drag ratios of these new con-
figurations as compared with those of the Tulin-Burkart design decrease
with increase in angle of attack.

3. The simplified equation developed for the cavity boundary stream-
line for arbitrary shapes is in good agreement with the more exact solu-
tion for the Tulin-Burkart Section and should be adequate for all low-
drag sections.

4. Low-drag hydrofoils developed from the linear theory cannot
operate at the design angle of attack because an upper surface cavity
will not form even for sections with zero thickness. The sections must
be operated at an angle of attack slightly greater than the design angle.


Langley Aeronautical Laboratory,
National Advisory Commttee for Aeronautics,
Lanley Fie~ld, Va., July ;2, 1957.






NACA RM L57Gla


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

5. Tachmindji, A. J., Morgan, W. B.. Miller, M. L., and Hecker, R.: The
Design and Performance of Supercavitating Propellers. Rep. C-807,
David Taylor Model Basin, Havy Dept., Feb. 1957.

4. Ripken, John F.: Experimental Studies of a Hydrofoil Designed for
Supercavitation. Project Rep. No. 52 (Contract N6onr-246 Task
Order VI), Univ. of Minnesota, St. Anthony Falls Hydraulic Lab.,
Sept. 1956.








NACA RM L57G11a


u m -4


U3 3 *

0 C S
1-1 -H 0) OJ

g E-

m


bD


0)



O
m
I'








O
a
'4 0
0


*rI

* '-4


0)
CH
K

VI
C


P /( a) 'umo q jo euMpzo IWuOTBuUiTPUON







NACA RM L57Glla 25








1------ C^^ q ______ ____ 0
H^^ / H ------- -g-'- U
H

P4
0 u 0

+ 0












40 co
4'
4-)

&4' 4




0 4
U,-~ *r-4










0e 0
4- -- .






.00
-'-0-/ --- ______ '0 K> 4'"r
\4-)---- ---- --- 4') *r










0

0


E4 E U 4 0:
'~~4 0 -




4-\rs G 4',








00

d H

~~4 CC i_ _
cm I
_____ __ H 4'*I
__ __ \ MI0 f





^ r -o -- ---H


---------- -- -- ____ /--- C0


0 0 0 0
Nr H '-aNX


LI


L






26 NACA RM L57Glla


100 i....i.*.. ..
IL. -JCONFIGURATIONS

A Tulin -Burkart f...
'w.
o B Three term

C Five term

Io
80 r... i
S- Note- Only the unshaded portion
is considered practical.
*1'~ (See text.) 1 .








) 50 *tLow-drag hydrofoils
operating at design

sCo





30 *- I ~ jl ~^l
,o~ 1, \1 e'''*^ ^ ^ ^ gl
40
.. '. *"; ^ ^ 1 : --^ ;.-;.i'











10 6T11"2' o o a^^^




o .1 .2 *h .5 .6 .7 .(
ev
30



10.

20 10





0 1 .2 .3 .7
CL


L/D and CL for low-drag hydrofoils.


Figure 3.- Variation of






IACA RM L57Glla


HYDROFOIL PLANE (Z)


AIRFOIL PLANE (Z= -Vi)


Figure 4.- The hydrofoil and equivalent airfoil planes.








NACA RM L57G11a


*-




hil
U~^
C,^"


I.


-31 ,


+0
d
d




P I









0.


00
CD
0 0
H
id- gi
42 >C



0
+2 di
-. *rc



U2

o



0 il

o b


V7
Cd

4-,


00

(U



.b( 0
0)




0


C-
0
U
cI


U;-







NACA RM L57Glla 29




Hydrofoil lower surface

Cavity upper surface
Angle of
attack.a

.2 deg..b



.1 -- ------3





-.1-
.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

x/c


(a) Tulin-Burkart.


.3-
o(,deg

.2- --6



r I
cc
01 .1 .2 .3 -4 .0 .7 .






(b) Three term.


.3-

o(, des
.2- ----6







o .1 .2 .3 *1.4 .5 .b .7 .8 .9 1.0
K/C


(c) Five term.


Figure 6.- Location of cavity upper surface for low-drag supercavitating
hydrofoils, CL d = 0.2.


NACA Langley Field. V.






I






















































































eIIl










UNNERSIry OF FLORWA



31
















'4

v




4



i 4 ,


4'















I It


It I




Ilk







1 41 'T

k'r'





iJ'l "'k












Y






t"Tt




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EP4J291N3_9TI599 INGEST_TIME 2012-03-02T22:59:49Z PACKAGE AA00009188_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES