Hydrodynamical investigation on the submerged hydrofoil

ARCHIEF

SYMBOLS

r

circulation

y = vortex distribution C = velocity

f = immersion, which is a vertical distance_measured

from still water level to the middle _{point of}

chord length = chord length

a= attack angle

p = density

g = acceleration due to gravity q = resultant velocity on the hydrofoil u = x-component of velocity on the hydrofoil

INTRODUCTION

Up to date, many theoretical _{treatments have}

been published of the submerged hydrofoil as a two dimensional hydrodynamical problem.

How-ever in these researches some theoretical weak points may be found as follows:

As far as the author is aware, no theoretical _treatments

have been seen of the submerged hydrofoil of arbitrary section.

It can not be said that the boundary condition ofzero normal velocity over the contour of the hydrofoil has been rigorously satisfied.

Hence the proper circulation around the hydrofoil has not been obtained and, consequently, the lift and wave

resist-Lab. v. Scheepsbouwkunde

Technische Hogeschool

Delft

tpSi 1e6 I TETSUO NISHIYAMA

HYDRODYNAMICAL

_{INVESTIGATION}

ON THE SUBMERGED

_HYDROFOIL

THE AUTHOR

studied Naval Architecture mainly at the department of Naval Architecture of Tokyo University in Tokyo, Japan from, 1944-1947. After several post-graduate courses, he has become an assistant professor in the department of Mechanical Engineering of Tohoku University in Sendai. His work is

con-nected with general hydromechanical _{problems in Naval Architecture, in} particular with wave resistance theory and with the submerged hydrofoil.

ance has been calculated only in rather rough

approxi-mate method.

Under these circumstances he has investigated

theoretically in detail the following_items:

Part I. A practical calculating _{method for} obtain-ing hydrodynamical characteristics of the submerged hydrofoil of arbitrary section.

Part II. A practical calCulating method for obtain-ing profile form of the submerged _hydro-, foil with a prescribed pressure

distribu-tion.

Part III. Hydrodynamical _{characteristics of the} submerged hydrofoil among stationary waves in relation to the problems of the tandem hydrofoil.

In this paper the first item will be

repo4tecl

exclusively.

PART I

1. FUNDAMENTAL EXPRESSION

Consider the two dimensional _{motion due to a} submerged hydrofoil placed in a uniform stream

including circulation r around the _{hydrofoil, the}

middle point of chord length of which _{being at a} depth f below the water level. Take the origin at the middle point of chord length, with ox horizontal A.S.N.E. Journal. August 1952 559

and oy vertically upward, and suppose the stream to be of velocity C in the negative direction of cm as shown in Figure 1.

We write the complex potential of the motion as

w=Cz+w; z=x+iy (1)

The condition to be satisfied at the free surface is

a2y.0

Re 1C2

-Fig--az az

After some reduction wo may be determined so as

to satisfy the condition (2), in the form

wo= F(k)e'kzdk k+Ko ip.F*(k)e- hz-2"dk. . (3)

where the limiting value tt--->0 is to be taken and the

asterisk denotes the conjugate complex quantity. Hence we write down (1) in a form

w=Cz+Az-n+Bozn+log z

op op 277 ir (4) where F(k)=CPC3' (ir An kn--/

ir

(n-1)! 27r k

(_i)fl co, k+KoiPF.(k) _{ne2k- fdk, ko=g/C,}

Bn= 0

Then, for the forces on the hydrofoil we have from

the Blasius' first formula

XiY=27rpi co k+Koi

_{o kK0ii,}

IlF*(k) F(k) k2e-krdkipCr (5) 560 A.S.N.E. Journal, August 19S0

Figure 1. Coordinate.

(2)

To complete the solution of the problem in any

given case, we have to determine the function F (k) so that the boundary condition of zero normal velocity is satisfied over the contour.

Now putting the pressure and velocity on the contour of the hydrofoil and in the undisturbed

stream (p, q) and (pun, C) respectively, we have

from Bernoulli's theorem

p+1/2pq2=p V2pC2 ₍₆₎

On the other hand, we have from (4)

dw dw*

q-=

dz dz (7)

Hence, from (6) and (7) we can calculate the pres-sure at any point over the hydrofoil.

2. TRANSFORMATION

Taking ox' along the chord length, which making a with ox axis, we have

zte'a=z; z'=xt+iy' (8)

Now we transform the hydrofoil of arbitrary section in z'-plane into a straight line (n=0, fl--271-) in C-plane by the function

z'//=Coeq+Ci+C2e1c+C3e2Ic+C4el+

-le-icl>1 (9)

where Ci, is a complex constant depending on the form of hydrofoil.

Further 77---)cocorresponds to the point at infinity

NISHIYAMA

HYDROFOILS

in the z'-plane. Hence the _{complex potential"' in E:.-plane is}

cr, , (I)'-w= A, e- ra

'e'("'"c+

_{B, et "1' 1:rye1("-ri'+ --(} 8e'") +C/e'a (n-nc (10) - I II U I/ nI nr:o where ry0=Cor, _{1y1=-1*C,C'-1,} (C12C.1-2-1-rC,"Cor-2), - -13f= ICo-' -1C2C(1-1-'+ 1/2r_{(C,2C-r-z-FrC,2C-1-2), -}

-SI=C,C-', 82=C,C-1-1/2C,2C-2 ₍₁₁₎

From the condition of zero normal velocity over the contour of the hydro-foil, we have

/=n cs

eh a+ Ar*e1ra+

_{ry* e-i ra=0}

_{. (12)}

27-with the additive term -C/Ceia for n=1. On the other hand, from Kutta's_condition

(dw d We have ct) Cie"

_{C(n-1)(-1)"-1-1- iE'{-1+:.n(-1)"}}

o 217 A, e-l'a (n+r) r /3,1 ( 1)"4r _I3,e"aIr

_{(n-r)r7(-1)n-r -0}

₍₁₄₎ r=1 II I _r-1 11=0

Accordingly, from (12) and (14), we obtain the infinite set of linear

equa-tions, from which the coefficient A. and B,. may be determined.

3. DETERMINATION OF COEFFICIENTS an, bn

In order to solve this equation, we put in the form

W 1

F(k) -iC/2ksin(a±/3)::: _(k

i,)"-'--(15)

(n-1)! k

where b,1_, is a complex constant.

T=4-kC/sin (a-140k, k= 1 +a, (KJ)_{+a, (K3)2-i-a3(1(003+ (16)} where

C0=ke3° _(16-1)

Substituting (15) and (16) in (4), _{and using the following relation}

jK+K±ip. Ke_,

"rc1K-Kr,"-"q

(17) K-Ko+ip,

-iS and S=2).re-2xf

n! _{(n-1)! 1} e+ - - +2-Kf - -2KoiCi(e2Kuf) rn= (2Kf)n+1 +2

{(2Kf)

we obtain Arz----ksin(cr+/9)C00'+'

(_i)r

1

-iCikosin(ced-igo) K'[K/)b*q,*+

(K01)13/*cl,+1*12!(Ko/Y1:02*(1r,..2*1 -r! (13) +2kqr_i*] (18)

Now we transform (12) and (14) in the form

1 r=n

C,,*e-ia *k2i8 i) "1 "Pn_r*e"br-1*

k0sin(a+13)

(i) " _{(K/) I r°[1+ (K/)bq+ (K002bici1,i+}

r=s 7: CO

-a r! (1(01)r ry,._ el" [2.4,1* -F (Kl)b*cIr*d- (K01)213/*c1,1*

- - -

2q,._,* 1(1=0 (12')

with the additive term

elakosin(a+13)for n=1 eia.:(n-1)C(-1)n-1k,sin(a+/3)o

1d-n8 (-1)n

(i) r+lbr ra (n+r) ( _1) n+r rtl n-a

(-01'

_{K,l)re'ra[(KOb*q,*F (K02b,*%.*+} -! r=1 r (nr) ry (-1)"-T =0 (14')

From the integral expression for q given in (17), it can readily be shown that the infinite determinant formed by the coefficient of b., b,, b2. . . on the left of (12') is convergent.

Although expansion in terms of other parameters may be more suitable for special ranges, it is convenient to assume that the coefficient can be ex-panded in power series of the quantity Ka. These expansions will be of the

form

b=4±b, (KJ) d-b.2(K3)2+

b,,b+b(K/)+b(K/)2+.

(19)

132=b2+b,, (K/) (K1)2+

Substituting in (12') and (14'), and collecting the various powers of Kol, the new coefficient may be found to any required stage.

From (12'), we have

1

for n=1, (C)*e-iaCoei a)

ksin(a+13) 2i81*--(-02 if3*e1a13*=0 (20-0) (-02 'S*eic131* +2(02 17,*e-laci 2( 02 ly0el°q*=0 (20-1) 2i*a2 (-021,80*elM,*+ (0'2 (q1b-1-2q0a1) (-02 ly0eia(c1th1o*±2q04+a1)

(03.,

(-03

+2 { -y

2! 27,e-"aq,* } (20-2)

1 for n=2,

C3*e-jaksin(ot+p) 2i8-0*(-0 1131*elab00 (-03 si3o*e2lab,o=0 (21-0)

2i82*al

(-02

i)3 2/3.*e'abli*+2(i)2 IY3*e-iaq0 (21-1) 2i82*a2 (-02 1/3i*ejah* (-032,80*e2iab*+ (j)2 1)':e° (q1b(0-F2hai)

(0". * _{(-0"2yoe-IN,*=0}

+2 {2, 'Y., e'ac11

2! (21-2)

562 A.S.N.E.Journal, August 1958

- +2qr_,

And also, from (14') we have el a (CoC,+2C-3C1+4C,

_{)+2i(P-81+282-383+....)}

kosin (a + ft) (i),e-1aboo(-1/3+2:1R.-3 '132+ ....) (i)e-2b01),0 (2 2(3,)-3 (22-0)

2i(-1-8,+2-3+....)a1(01e_iab,1 (V30+2 'Pt-3 I/32+....)

_(i)3e-2iab,, (2 2p0-3 2p, +

_

_{+2(_i)ei0go. ('yo_1y2±2 ) =0 ... (22-1)} 2i ( _{8 , +28,-383+ ) a, (i)2e' °3,2 (_1/3+2}

+2 V30-3 1/32+ )

(i) 3 e-2iabi2 (2 2/33 2/3,+ ) i) 2ei a (2a,q,)*+q,*boo*) ('70-1y2+2

+ 22 'aqi* 27+27L-27+ _{) =0}

2! (22-2)

From (20-0) and (21-0), it can be readily shown that the equation

(22-0) is

automatically satisfied.

Substituting (20-1) and (21-1) in (22-1), we obtain

(-0=eicrq*iy (i)2e-jaq17* _{. (23-1)}

Similarly, from (20-2), (21-2) and (22-2)

a2= (i)2e-1a(2q)a1+q,b)17*(i)=e1a(2q0*a1+q,*)1y0}-3e-21acheyi*_2 _{e2ia ,*(27._2 270)}

(23-2)

2! 2!

and so on.

Hence from (20) and (21) the coefficient b may be readily evaluated.

4. LIFT, WAVE RESISTANCE AND PRESSURE DISTRIBUTION

Now putting the lift coefficient and wave resistance coefficient in the form

respectively,

CY/1/2pC=1, X/ 1/2pC2/ we obtain from (5)

Ca=87k ksin(a+/3) 471(2sin2(a+P) [41.C.2(K/)r,±2k _{(K1)2(bo-I-b(,*)r,} (K1)113,,b,,*r,+2k (b,-i-b*)r,±2k(b+b*)r,1

(24)

C.=47ksin2(a+/3) [4k= (KJ) S+2k (K4) 2 (b)-1-b(*) S

±(K03/ bobo* +2k (bm+b,*) S +2k_{(b,+b*)S}-+ (25)}

From (6) and (7), the _{pressure on the contour of hydrofoil is}

1/2C2

C Xda ddY;

(6') where

1 I/2k/ sin (a+,80) =k+kesin (a+Po

[ksin(a -1-430)

2(K01)kr (1V) 2 (bi*ribw,*S)

2kk, (K(1) 2 iriSin(a+/31) +S cos(a+/3,)}--2kk_{(K)/)=.irlsin(a+,130$)} +S cos(a+,13-'-) + . _{kocos(a+/3.)[2(KOkS± (K(,/) 2 (b0(12''S+b002*r1)}

2kk1 (K01)1 r,COS (a+,81) S sin(a+fl, ) 2kk0 (K,,/)2 ricos (a-1-13--$)

s sin(a+,00$) and _{b0=b001-Fib002}

A.S.N.E. Journal, August 1958 ₅₆₃

NISHIYAMA

HYDROFOILS NISHIYAMA

5. FOURIER ANALYSIS From (9) we have

x'// = C11+ (C,Icose Csine) +. .. .

(9')

y'/1= (CcoseC1sin) +C12+ (C;nsine+Ccose)+ where Cu=Cii -FiCn

Now, analyzing the section of hydrofoil by Fourier series, the following equations are obtained:

UD

x'/ I= 1/2Cos$ 371/t= Mncos Nsin nE (26)

Comparing (9') with (26), we have

C 11:=1V1(19 CO2 + C.22=M1/C32= 142, C42=M3/

C21= N1, C31= 1\12, C11= N:1, (27)

then, from (9')

x'//Nrf k /1= 2 (C,,,,sine.+Colcose) +C11 (28-1)

x'* / 1+ y'a= 2 (CO2coseC,sine) +C12

Inserting (26) in (28-1), differentiating by and putting E=0, we obtain

cip C =0,Cfrz = ½nM,C22 =MI 1/2EnMn 1 (/) CO C12=MO/C111= 1/2 (1/2 +:nN),C=N,+1/2(1/2+EnN) (28-2) 1 1 hence, from (16-1) U) CO GO OD

k=1/2{ (1/2+.`:nN1)2'+ (-`:r1M.)2}1/2 flo=tan-1 I..rilani (1/2+EnN)I (16-2)

1 I 1

1

6. SPECIAL CASES

Now we consider some special cases from the practical interest.

Thin hydrofoil

Putting the coefficient 1\1,1, depending only on the thickness, zero, we can obtain the characteristics of the so-called thin hydrofoil having only a camber, from the corresponding expressions. In this place our interest in confined to the vortex distribution equivalent to the thin hydrofoil.

From the definition of the vortex distribution wehave

7(0 (29)

E>0

Taking u=q into account, from (6) and (6') we obtain

CO

(1/2sina+ I 5_.'nM I cosa)

(e) =C CO G (29')

1/4Sin2C+ (EriMnsin nf,)2

where G=2k+2cos

2k0sin (a +13) [2 (K1) krocose+ (K002 (130,* rboo.,*S)cose

-F2kk,(K01) risin(a+fl,) -FS cos(a+/3, }-cose

+2kko(K0/)'.1r1sin (a +/3) +S cos (a- - flo)

....]

2k0cos(a+13(4) [2 (Ko/) k S cose+ (K/) 2 (b00./*S+boo.,*ri)cos

2kk1 (K/) 2/r,cos(a+131) S sin (a+ /3,)cose

2kk4(K/) 2/ ricos(a+130) S sin (a+) }cos2e+ Submerged flat plate and circular cylinder

CC.,=1/4 otherwise C=O

C0=1 otherwise Co=0

The former corresponds to the submerged plane hydrofoil(2) and the latter to the submerged circular cylinder.")

Characteristics in an infinite fluid

It can be easily seen that if we take the limiting case f)00, the correspond-ing expressions are in good accordance with Prof. Moriya's results(5) which is familiar in Aerodynamics.

7. EXAMPLES OF PRACTICAL APPLICATION

In a practical calculation, only the following procedure is necessary.

First, the coefficient M, N,, should be determined by Fourier analysis of a given hydrofoil. Hence from (28) the coefficient _{is obtained.}

Second, from a given speed and immersion, _{a value of Kf should be} de-termined, from which r,, and S are obtained by using the _{following Table.}

Hence, from (23) a value of k is evaluated.

3. Lastly, substituting C. and k in _{(24), (25) and} (6'), the lift, wave resistance and _pressure dis-tribution are readily calculated.

As a numerical example, we take an NACA 4412 section. Curves are given showing the lift coefficient in Figure 2 and wave resistance coefficient in Figure 3 for K0f=3, 1, 0.4, 0.1, 0.02 and 0, and _{f /=1; it is} readily seen that the inclination of lift _coefficient and zero-lift angle varies with Froude number, and wave resistance coefficient, in general, is proportion-al to the square of attack angle for _{every Froude}

number.

Curves are given showing the _pressure distribu-tion in Figure 4 for a=5", f//=1 and _{KJ=3, 1 and} 0.1; in which the area of curves on the left side is proportional to lift and the difference of area of both curves, full and dotted line, on the right side, to wave resistance. Hence _{we can see that for} rela-tively high Froude number the possibility of

cavita-tion occurring on the back of hydrofoil is rather

diminished.

Now we compare with the results _already pub-lished. First, curves are shown giving the wave

resistance calculated by Ausman's 4 _{and author's'")}

approximate method in Figure 5; from which we

can see that the former is much larger and the lat-ter is a little smaller than the present exact method. But at small attack angle the latter seems to be practically satisfying. The reason for overestimation

of Ausman's method _{seems to lie in the neglect} of variation of circulation _{or lift by effect of}

im-mersion.

Curves are shown giving the lift and wave resist-ance in the same condition as Ausman's

experi-ments' in Figure 6. And also by adjusting_the cir-culation such that the both lift., in _{an infinite fluid} experimental and theoretical, takes a same value at same attack angle, the lift and wave resistance is corrected. Both corrected curves are generally in

Figure 2. Lift. (Dotted line shows the _{corresponding value} in an infinite fluid) _

-KJ S ro _r, r2

- -

_{r:1 r.,} 0.06 5.5726 10.5176 88.6281 1315.4653 31407.5809 966959.7403 0.1 5.1442 6.3456 36.3456 311.3456 _{9363.4566 83061.3456 2033061.3456} 02 4.2117 2.3595 11.1095 48.6095 314.2345 2892.3595 36081.5934 0.4 2.8231 0.0392 2.8517 8.3204 26.8751 114.7657 645.7707 0.6 1.8924 -0.6377 0.8900 2.7418 6.2927 9.3310 69.1135 0.8 1.2685 -0.8307 0.1880 1.0638 2.4676 5.6719 15.1132 1.0 0.8503 -0.8409 -0.0909 0.4091 1.0341 2.1591 _4.7835 2.0 0.1150 -0.4689 -0.1564 -0.0626 -0.0079 _{0.0389 0.0916} 3.0 0.0155 -0.2595 -0.0650 -0.0280 -0.0141 -0.0065 -0.0008 4.0 0.0021 -0.1705 -0.0298 -0.0103 -0.0059 -0.0028 -0.0016 NISHIYAMA HYDROFOILS

1

0.10

--.

17(

Figure 3. Wave Resistance.

good accordance with the experimental in a larger immersion than f/1=1.5, but in the extremely small

immersion the difference is relatively large. This

-reason may he interpreted to lie in the occurrence

of the hydraulic jump on the back of hydrofoil;

hence the experimental values of lift and wave re-sistance are smaller and larger respectively, due to hydraulic loss, than the theoretical.

Lastly curves are shown giving the vortex distri-bution equivalent to the fiat plate for KS=0.8, 0.4, 0.1 and 0 in Figure 7.

_ approximate (by ausman

approximat e by a.titor

exact

by autitor)

005

Cur

566 P.S.N.E. Journal, August 1958

05

Figure 4. Pressure Distribution. (Dotted line shows the

corresponding value in an infinite fluid)

Kot =0.4 0.8 0.1 0

HYDROFOILS NISHIYAMA

--1.

Figure 7. Vortex distribution (f/1=1.0). (Dotted line

shows the corresponding value in an infinite fluid)

0 50 10°

Figure 5. Comparison (K,f=0.10, 1/f=1.0)

SILVERSTEIN

8. CONCLUSION

A solution is given for the two dimensional _wave motion due to a submerged hydrofoil in _{a uniform} stream, taking fully into account the condition at the surface of the hydrofoil. Expressions for the lift, wave resistance and pressure distribution on the hydrofoil are obtained in the form of infinite series

in ascending powers of a certain parameter. Nu-merical calculations are made from these and com-pared with the approximate solutions and experi-mental results.

REFERENCES

1. Goursat-Hedrick. "Mathematical analysis," Vol. 1.

The lift, wave-making resistance and pitching

moment of a hydrofoil moving beneath a free

sur-face have been explored theoretically and

experi-mentally. None of these aspects is completely

understood today. This paper by Professor

Nishi-2.0

1.0

DISCUSSION

"Hydrodynamical Investigations on the Submerged

Hydrofoil"

by Tetsuo Nishiyama

By Dr. B. Silverstein, Ship Design Division, Bureau of Ships

measured

( by ausman)

calculated

calculated ( corrected)

T. Nishiyama, "Effect of immersion _{on the characteristics}

of the submerged hydrofoil," (2nd-report), Jour. of Soc. of V.A. Japan, Vol. 95, 1954.

T. H. _{Havelock. "The forces on a circular cylinder}

sub-merged in a uniform stream," Proc. Roy. _{Soc.. Vol. 157,}

1936.

J. S. Amman, "Experimental investigation of the influence

of submergence depth upon the wave-making resistance

of an hydrofoil," Master of Science thesis, Univ. of Calif.,

1952.

T. Moriya. "A calculating method for obtaining charac-teristics of aerofoil of arbitrary section." Jour. of Soc. of

Acre. Eng.. Vol. 5, No. 33, 1937.

T. Nishiyama, "Effect of immersion on the characteristics of the submerged hydrofoil," (3rd-report), Jour. of Soc.

of N.A. Japan. Vol. 96, 1955.

yama is one link toward the development of a suitable theory.

In the future, there will be more papers on this subject as the complexities have attracted _many

mathematicians. In the past, Kochin, Keldysch.

HYDROFOILS

1.0

_2.0

HYDROFOILS SILVERSTEIN

Lavrientiev, and others have set up and solved cer-tain simplified mathematical models of the hydro-foil problem. It is interesting to note the manner in which the solutions of other investigators were de-veloped. J.

P. Breslin, in a recent paper in the

Journal of Ship Research, utilized the hydrodynam-ic theory of wave-making resistance of ships, de-rived by T. A. Havelock. This theory was applied to a hydrofoil of finite span which checked with the definitive, earlier studies of Y. T. Wu. A. G. Strand-hagen and G. R. Seikel, in a recent paper on the

lift and wave drag two-dimensional hydrofoils

utilized the concept of a "substitutional" vortex re-placing the foil. A comparison of Professor Nishi-yama's approach to Kochin's might be illuminating insofar as it illustrates two methods of attack.

Kochin, Kibel and Roze rigorously formulated the problem of steady, two-dimensional motion of a

body near a free surface. This is readily derived

utilizing 0 and the velocity potential and stream function, respectively (In equation (1) of this

paper, w=0H-ict). On all solid, fixed boundaries, such as the surface of the body or hydrofoil, there can be no fluid flow into the body. A second condi-tion is that the free surface is a streamline; that is, a particle that is initially on the free surface stays at the free surface. Third, the pressure is constant on the free surface. The fourth condition is that far

ahead of the hydrofoil there is no wave motion, though waves may exist far aft. This last boundary condition is based on the observed physical behavior of bodies, such as ships, moving on or near the water's surface. It might be noted the symbol "it" of Professor Nishiyama's equations (2), (3), (5), etc., is employed as a mathematical artifice of satisfying this condition. It is often called, as utilized by Ray-leigh, a "fictitious viscosity." Fictitious, because in all developments, after serving its purpose, p. is set equal to zero.

These four conditions are applied to Laplace's equation, a second-order, partial differential equa-tion; both 0 and ,p satisfy it. Laplace's equation is simply a mathematical expression of the continuity of the fluid; no liquid is created or destroyed, ex-clusive of sources or sinks.

Appropriate mathematical expressions are then

derived by using the well-known perturbation

method. This assumes that 0, II, and the surface wave height, 7 can be expanded in a power series of

a small parameter, r, which is to be selected such that as E approaches zero, the motion approaches pure uniform flow. After substitution into the four boundary conditions, mathematical expressions for the linearized and non-lihear problems arise. One such expression, where p.=0, is equation (2) of this paper. This checks the linearized free surface condi-tion of Kochin, Kibel and Roze.

The simplest sort of representation of a hydrofoil 568 A.S.N.E. Journal, August 1958

is a single vortex. For bodies that do not depend primarily on lift, such as displacement ships and submarines, sources and sinks are extremely useful as aids in the mathematical description of character-istics such as wave-making resistance and motions. But as proved so important to the rational develop-ment of aerodynamics, a distribution of vortices best simulates lifting surfaces, such as wings and

propel-lers. For a vortex of intensity I- at z=ih, the com-plex velocity potential is

if'

w= (z±ih)±g(z),

where the first term of w applies to an infinite fluid and g (z) is a function that is to be determined so that w satisfies the boundary conditions at the free surface. After suitable manipulations, the following relation for the velocity potential is obtained:

iF

zih

if' ciK t

log"z+ih r

eK,7

tih

CID

The Blasius formulas for lift, Y, drag, X, and moment, L, of any body in motion in a perfect fluid can be readily developed. X, for example, is the sum of the pressure components parallel to the line of motion. By using Bernoulli's theorem and the prop-erty that the body is impermeable, the Blasius form-ulas may be written in terms of the complex ve-locity potential as follows:

p2 .0.( ddl) -dz,

L=Re[i-

(t-) 2

z dz].

For a single vortex, of intensity r, located at

z= ih, the following results are obtained:

pr2 Y=Pre-4,h gh L=p1-2 c- e CY) e. Where Ei1(z)=.1du . U

Tables of Ei, (z), the exponential integral, may be found in Jahnke and Emde.

These formulas are the results for the simplified model of replacing the hydrofoil by a single vortex of unspecified circulation l'. The proper determina-tion of the circuladetermina-tion is needed.

..2gh

Ei,(2gh),

7re-SILVERSTEIN _HYDROFOILS

Kochin attacked this particular deficiency in the theory by developing his famous H function. This has the advantage that the free surface boundary conditions are incorporated within the H function. Thus, if the hydrofoil is represented by a distribu-tion of vortices, one need only consider the contri-bution of these vortices to the velocity potential.

where S is the area enclosed by the two-dimensional

hydrofoil.

The standard technique of transforming the lift-ing surface into a circle was not applicable. Map-ping functions of the hydrofoil and the free surface into a circle would be extremely complicated and, at present, have not been developed. Kochin, there-fore, omitted the free surface and by fairly

exten-sive computations inverted his formula and

ob-tained the velocity potential, w, in terms of the H function. In this expression, Kochin introduced the Kutta-Joukowsky condition; this is the assumption that the trailing edge of the hydrofoil is a stagna-tion point (zero velocity). This resulted in an integral equation relating H and F. Usinga mapping

function of an airfoil into a circle, H can be com-puted and substituted into the integral equation in

order to derive an approximate value for _{P. An}

iteration procedure could be used to further refine the accuracy. This method works well for a flat plate, but is not satisfactory for general sections.

Professor Nishiyama's approach is similar in prin-ciple. The point of departure appears to be primari-ly in the order in which the boundary conditions are applied, and in the fact that equation (2), the differential form of the free surface condition, is

transformed into equation (3), an integral form.

This is substituted into the Blasius' formulas. The hydrofoil is mapped into a slit and the velocity po-tential reformulated. The other boundary condi-tions are imposed, including _{zero normal velocity}

00 pK,, dA Y=.--per-ppgS PIdX1H(A)12-1- - )12, X 0 -CAD d (X) =.Cr/ dzw -dz

e,

where 1 is the contour of the hydrofoil.

This results in a different representation of the Blasius formulas:

X= pKIH(K)12,

and

over the hydrofoil (an impermeable body), and the Kutta-Joukowsky condition at the trailing edge. Professor Nishiyama demonstrates considerable in-genuity in the collection and evaluation of the con-stants after they have been affected by each boun-dary condition. Perhaps a weak point in this devel-opment was the use of evaluation of lan in a power series. The usefulness of the result thus depends on a rapid convergence of this series, in the para-meters K1. However, the final, reasonable correla-tion of this theory with Ausman's experimental data

(Figure 6) is encouraging.

It might be pointed out that studies, such as this by Professor Nishiyama, serve a useful and practical end. The naval architect, interested in designing the foils of a hydrofoil craft, would not use the results of this paper for a design procedure. He is interested in cavitation effects, effect of the intersection of the foil with the strut, and in a finite aspect ratio foil.

Very often, however, the results of studies of a mathematical model of reality shed startling light and insight, on physical phenomena.

Naval engineers might well be alerted to the in-flux of mathematicians into the field of hydrody-namics. It was primarily due to the efforts of ap-plied mathematicians and theory-minded engineers that the field of aerodynamics developedso rapidly.

Mathematical approaches to ship motion and

wave-making resistance of ships and hydrofoils are

al-ready paying dividends. A sound theory is an

ex-cellent guide for conducting, analyzing, and under-standing experiments.

96 4010 .10-47.1.;,..teliff-:, _{T..1"4"".`"P"' T:1/4 .7.144rv"} _ .

SYMBOLS Ca = lift coefficient (L/1/2pC21)

C = wave-making resistance coefficient (R/1/2pC2/) = chord length

f = immersion depth, which is defined as a vertical

distance from still water level to the

mid-point of chord length a = angle of attack

g = gravitational acceleration

K = g/C2

Subscript means that the matter is not under the effect of the free water surface

Subscript , means that the matter is under the effect of the free water surface

INTRODUCTION

The pressure distribution is closely related to the transition and separation of the boundary layer, and the inception of cavitation over the back of a hydro-foil. Therefore, in order to research the suitable pro-file form of the hydrofoil for a given condition of operation, it is of primary importance to establish

TETSUO NISHIYAMA

HYDRODYNAMIC INVESTIGATION

OF THE SUBMERGED HYDROFOIL

THE AUTHOR

studied Naval Architecture mainly at the department of Naval Architecture of Tokyo University, Tokyo, Japan from 1944-1941. After several

post-grad-uate courses, he has become an assistant professor in the department of

Mechanical Engineering of Tohoku University in Sendai. His work is con-nected with general hydromechanical problems in Naval Architecture, in

particular with wave resistance theory and with the submergedhydrofoil.

PART II

A method for obtaining the profile form of the submerged hydrofoil with_a prescribed pressure distribution.

a theoretical method for obtaining the profile form of the hydrofoil with a prescribed pressure distribu-tion under a given condidistribu-tion.

From the standpoint of thin wing theory, War-ren proposed a method for obtaining the profile form, in particular delaying the _{occurrence of} cav-itation. However, this method neglected the exist-ence of the free water surface which has a unique and important effect.

Generally the existence of the free water surface decreases the lift of the submerged hydrofoil for normal speeds. Hence, if we apply the profile form designed by Warren's method to the submerged hydrofoil, we meet inevitably with the fact that the lift coefficient of the submerged hydrofoil is_smaller than the designed lift coefficient, even if the occur-rence of cavitation might have been delayed. This shows that Warren's method is on the dangerous side.

Under these circumstances, taking the _existence of the free water surface into account, a method for

obtaining the profile form of the submerged hydro-foil with a prescribed pressure distribution is pro-posed from the standpoint of the theory of the sub-merged hydrofoil of arbitrary section. At the same time the suitability of the types of the pressure

dis-tribution is discussed for the submerged hydrofoil.

BASIC IDEA

As the pressure distribution can be easily con-verted into the velocity distribution from Bernoul-li's theorem, we hereinunder use only the latter.

It is rather difficult to treat the prescribed veloci-ty distribution in that condition under the effect of the free water surface; therefore, after converting

the prescribed velocity distribution into the cor-responding distribution in an infinite fluid (without the effect of the free water surface), we calculate the profile form of the submerged hydrofoil from

that corresponding distribution by the already

known method."

If the velocity distribution in the case where a hydrofoil is affected by the free water surface and

in the other case where it is not affected by the For a hydrofoil of the plate section

Lift coefficient:

where

with and

not affected by the free water surface:

C.=2rsinn (2)

Velocity distribution: v/C7---cos+cot :11 sinn

2

Coordinate: x/1= 1,4cos7); 27>71=-0 (4)

For a hydrofoil of the plate section affected by the free water surface:

Lift coefficient: Cu,..=---27sina,[k 1/2K/sina { k3r.

+1/2K/kr,sinne+.1/4K213 (1/2k3s sin2a-1/4r,

k3rr1sin3a,k3r,s sinnecos a,,)

± - - - 1

(5)

where _{k= 1+ 1/2a 1 (Kul) + 1/4a2 (KJ) 2 + - - (6)}

(3)

a1= (rsina+s cosa,) (rsinne+s cosae) 2 1/2r, ( 1 + cos 2(c)+ 1/2s sin 2n, s=27e-3Kor -= n! {-(2Kf)(n-1)! 1 n (2Kf)" +2 + - +2Kf e-3Ko1Ci(e3.Kof)} (7) Velocity distribution: 2 11 (C, sin nn+C,, ,cos +k v,./C= _{sin n} Sinn, (8) 1 C = 1/2cotn,-1/4kK1(rcosa, s sina,),+ K0212s+ -16 1

C, =

k K, 3/3 (r,sin2a,± s cos2a,)+ -16

C, ;=1/21/4k Kl(rsinn,± s cos2a,) +

-16 1 kK2/2(r,cos2n,s cos2a,) + -= 16

same (i.e. in an infinite fluid) are denoted by v.. and v respectively, and the quantity of the change in the .velocity distribution due to the effect of the free

water surface is denoted by Ave, we have the

re-lation

ve/C=v/C-1-.AvdC (1) where C is the advancing speed of the hydrofoil. Since the lefthand side of (1) is given, if the second term of the righthand side can be obtained, it is

possible to convert it into the velocity distribution in an infinite fluid, and consequently to calculate

the profile form of the hydrofoil by the known method.. At the first stage of calculation, v,/C cannot be obtained directly since the profile form of the hydrofoil is still not known. However it can be obtained by successive approximations.

FIRST APPROXIMATION

For the first stage of approximation we assume that Av,./C is equal to that of the plate section. Then we can use the theoretical solution 4 for the hydro-foil of the plate section.

17 .` '

In general there is the relation

A sina

' (9)

cosa

between an area A surrounded by the pressure distribution curve in the usual non-dimensional form in wing theory, and the lift coefficient C. and wave-making resistance coefficient Since the angle of attack is small, it is given with sufficient accuracy by

C=A (9')

When the immersion depth and the speed of the hydrofoil are given, and

a.,. are obtained from (2), (5) and (9'), respectively. Substituting a and a,. into (3) and (8), respectively, we may obtain Av,/C. Then v,./C can be ob-tained from (1), and consequently the profile form of the hydrofoil can be

calculated.

SECOND APPROXIMATION

In order to increase the accuracy of calculation, it is necessary to calculate Ave/C for the profile form of the hydrofoil obtained by the first approxima-tion. Then the theoretical solution 1 for the submerged hydrofoil of an ar-bitrary section can be used for this purpose.

For a hydrofoil of an arbitrary section not affected by the free water

sur-face:

Lift coefficient: C.,=87ksin (a-÷ [3)) (10)

Velocity distribution:

v/C=

2k; sin (,r+3) +sin (a.-1-13y)

nasinny +

nbcosny)-where

_k=

)'-f- (1/2+ nb)1),,; /3=tan I I na/ (1/2-1- nb)1

co cn

Coordinate: x/1= V,cosy, y/1=.`:acosny b,,sinnn; (12)

27- 0 1

For a hydrofoil of an arbitrary section affected by the free water surface:

Lift coefficient: (a,+/3) 41c2sin2(a,+/3) [4101{.1r

+8k21<2/'1.,k ,sin(0,+/3) 4kK21-'r,lk,cos(2a+/31-fl,) -(13) where

k=l+a, (K1) +a,(K/)-1- - - - (14)

with

al= 2k1 rsin(a,+/3) +s cos (a,+,8)

a,= 4k.21 rsin(a,+13) -1-s cos ((l'e±f3) }.2

2kk1-{ r1cos(/31 /3) s sin (fl,

+21.,1 kk1cos(13+fl1 +Ire) kcos (2a,+2,0) kk ,sin ([3 + +2a,,) ksin(2a,+2/31) + k2sin (ae+132) +ksin (a,+ p)

s k2cos(a,+,8,) kcos(a±/3)

A.S.N.E. Journal, November 1958 665

HYDROFOILS 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 and and

The coefficients an and b can be determined by

carrying out the Fourier analysis for the profile form of the hydrofoil obtained by the first approxi-mation. The said form in the second approximation can be calculated by the use of

and b and by

the same procedure as that of the first approxima-tion.

Figure 1. Three types of the prescribed pressure

distribu-tion with Ce=-06

666 A.S.N.E. Journal, November 1958

k2e1/1,= b,+1/2(1/2+nb)-i-i(a1.71/2..na)

1 1

Velocity distribution:

(15) /1/4sin'n+ ( >.:nansinm1+ >.:nb0cosn77)

1 1

where

I/2ksin (a(,+,0) =k +1sin (re-Fp-77) [ksin(a,,d-p)

2KdkrK212 i*r,b(1 2*s) 2kk,K2/=/ risin (0e-f-$1)

+s cos(n,+,() t(-2kkK02/'{ risin (cc-EX-0

-Fs cos (rte+ /3

- - -

_{kcos (a,+ /30 70 [2K0iks} -FK212 (b i*s+b, 2*ri) 2kk,K212 r,cos (a, +PI)

s sin (a,+,8, ) 2k1c01<212-{ r1cos(a,+,31-77)

s sin (oze-F/3,, 77)

- - - ]

1 +2kisin(ae-}-,81) b0.1*= k2eos (2a, +/30-0.,) A) _x sin (a,..-0)) 1 +2k,sin (ae+/31) k2sin(2a,+A.-F/32)x sin (ae+ 8.)

NISHIYAMA

NUMERICAL EXAMPLES

The pressure distribution, aimed at the lift coeffi-cient C.,.=0.6 is shown in Figure 1 (a). It has a

min-imum immediately after the leading edge and a gradual rise to the trailing edge. From this pressure distribution the profile form was calculated under

the condition:

Immersion depth f//=1.0

Froude number C/Vg1=3.16 _(K/=0.1) and shown in Figure 2; the difference between the first and second approximation is that in the latter the thickness is greater in the fore part and smaller

in the after part and the camber is smaller in the latter, but these amounts are very small.

Now we calculate the profile form ignoring the effect of the free water surface, as per Warren's method. From the pressure distribution shown in

Figure 1(a), the profile form obtained is as indi-cated by the dotted line in Figure 2; the feature of which is that the thickness is smaller but the cam-ber is almost constant.

When this profile form is usedas that of the sub-merged hydrofoil in the above mentioned condition, the lift coefficient can be calculated by applying the theory '2 _{of the submerged hydrofoil of arbitrary} section; we obtain the lift coefficient Cne=0.4665 which is approximately 20 percent less than the des-ignated lift coefficient Cue=0.6. This decrement is exclusively due to the effect of the free water sur-face. Hence this fact tells the importance of consid-ering the existence of the free water surface in the design of the profile form of a submerged hydrofoil.

A144, 411!",

NISHIYAMA

2nd approximation

with free surface

1st approximation without free surface

0.2 0 4 0.6 0.8 10

Figure 2. Comparison between first and second approxi-mations of the profile form corresponding to the prescribed pressure distribution (a) in Figure 1

(K.1=0.1 f/1=1.0)

SUITABILITY OF THE TYPES OF PRESSURE DISTRIBUTION

It is an important matter in the design of the

pro-file form to examine the suitability of the types of the pressure distribution. When we choose the pres-sure distribution for designing the submerged hy-drofoil, the following items should be taken into

account:

make the minimum pressure as small as possible, in

order to avoid the occurrence of cavitation.

make the pressure gradient immediately before the

trailing edge as loose as possible, in order to decrease the form resistance due to the separation of the boun-dary layer.

make the position of the minimum pressure back

toward the trailing edge, in order to decrease the

fric-tional resistance.

make the wave disturbance on the water surface as

small as possible, in order to decrease the

wave-mak-ing resistance.

Now we examine the relation between the wave-making resistance and the types of _pressure distri-bution. The three types of pressure distribution on

the back with the designed lift coefficient

HYDROFOILS

will be considered and these are shown in Figure

1 by (a) (pressure rise), (b) (pressure constant) and (c) (pressure descent). From these the profile forms are obtained under the condition

Immersion depth f/1= 1.0

Froude number C/Vgi=3.16 (K1=0.1)

and shown in Figure 3.

For these three profile forms the

wave-making

resistance coefficients can be calculated as follows:

The difference between these three wave-making resistance coefficients is negligibly small; hence it

is not too much to say that, so far as the wave-making resistance is concerned, the type of pressure distribution has little significance as long as the lift coefficient is maintained the same.

From the above results the types of pressure dis-tribution should be selected, taking items [1] [2] [3] into account. a 0.2 04 0.6 Camber distribution 0.2 0.4 06 0.8

Figure 3. Profile form corresponding to the three

pre-scribed pressure distributions (a) (b) stud (c) in Figure 1 (IC.1=0.1 f/I=1.0)

A.S.N.E. Journal. November 1958 ₆₆₇ 10

0.8 1.0

a

Ct, _{0.0163 0.0163 0.0165}

HYDROFOILS _NISHIYAMA CONCLUSIONS

The main problems treated in this paper are as

follows:

Applying the theory of the submerged hydrofoil of an

arbitrary section, a method for obtaining the profile form of the submerged hydrofoil with a prescribed

pressure is presented.

It is shown that neglecting the effect of free water sur-face, such as Warren's method, leads to the dangerous side in the design of the profile form of the submerged

hydrofoil.

The suitability of the types of pressure distribution is examined for the submerged hydrofoil.

668 A.S.N.E. Journal. November 1958

DISCUSSION

Review of Professor Nishiyama's Paper

(Part II) There are two closely-related problems in airfoil (or hydrofoil) theory. The first is the determination of the characteristics (lift, drag, moment, pressure distribution and flow pattern) of any selected foil. The second is the so-called inverse problem, where the pressure distribution is given and the foil shape is to be determined.

The first problem was solved for an arbitrary two-dimensional foil near a free surface, by Professor Nishiyama in a recent paper published in the Aug-ust 1958 JOURNAL. The second problem is herein presented and represents a logical extension of the earlier study.

Professor Nishiyama's procedure is straightfor-ward, although, as in his previous work, consider-able developmental work is omitted. A method of successive approximations is outlined such that the accuracy of determining the foil shape improves with each step. The first approximation is to assume a flat plate in an infinite fluid and to compare its velocity distribution with that of a flat plate near a free surface.

The velocity distribution of the plate near a free

surface is then assumed to arise from some

un-known foil of a certain shape in an infinite fluid. This unknown shape may then be calculated by the procedure given by Warren.

If this latter reference is not readily available, the process is in many of the textbooks on aerodynam-ics, such as Milne-Thomson's "Theoretical Aerody-namics," Second Edition, page 148. This method in-volves the separation of the velocity, or its equiva-lent pressure, distribution into two functions, one

odd and the other even. The odd function

deter-REFERENCES

C. H. Warren, "A Theoretical Approach to the Design of Hydrofoil," Aeronautical Research Council (British),

Re-ports and Memoranda No. 2836, Sept. 1946.

T. Nishiyama, "Hydrodynamical Investigation of the

Sub-merged hydrofoil," JOURNAL OF THE AMERICAN SOCIETY OF

NAVAL ENGINEERS, Aug., 1958.

H. Kikuchi, "A Method for Obtaining Profile Form with a Prescribed Pressure Distribution," Rep. Inst. High

Speed Mechanics. Tohoku Univ. 4 1954.

T. Nishiyama, "Effect of Immersion on the Characteristics

of the Submerged Hydrofoil (2nd report), Jour. of Soc. of

Nay. Arch. of Japan vol. 95, Aug. 1953.

on Hydrofoils

mines the camber line function, Yr., and the even function, the thickness function, Y. As developed by Milne-Thomson, the profile can be determined by two relations, Y, the equation of the upper side of the profile and Y1, the equation of the lower side. We then find that

Y,Y,.-EY, and Y, =Y, Y,.

Professor Nishiyama uses the same procedure for his second, and more accurate, approximation. Here he compares the velocity distribution of the profile, developed in the first approximation, in an infinite fluid and near a free surface. In this he utilizes the results of his earlier paper. Then the procedure for obtaining the profile is indicated to be the same as for the first approximation.

In the practical design of hydrofoils, it is neces-sary to determine the lift, drag and cavitation char-acteristics. The usual procedure is to select a pro-file from the many series available in the literature, such as presented in NACA Report No. 824. The profile is picked so as to have the desired character-istics. The more desirable approach is the one given by Professor Nishiyama in this paper. A pressure distribution can be quite readily selected that will yield the desired characteristics; the determination of the foil shape from this should yield a more sat-isfactory hydrofoil than one selected from an arbi-trary series of foils. It should be noted, however, that an extension of this paper to foils of finite as-pect ratio would provide information much nearer the needs of the designer.

In this, and in the earlier paper, Professor Nishi-yama develops tools and methods that should even-tually prove quite useful to the hydrofoil designer.

HYDRODYNAARCAL

_{INVESTIGATION ON}

THE SUBMERGED

_{HYDROFOIL PART}

_III

THE AUTHOR

studied Naval Architecture mainly at the department of_{Naval Architecture}

of Tokyo University, Tokyo, Japan from 1944-47._{After several post-graduate} courses, he has become an assistant_{professor in the department of}

Mechanical

Engineering of Tohoku University in Sendai. His work is connected with general hydromechanical problems in Naval Architecture, in particular with_{wave resistance theory and with the}

submerged hydrofoil.

SYMBOLS = wave length (27/K0 ).

= amplitude of stationary _wave. phase of stationary _wave.

2p or t = chord length of _{a plane hydrofoil.}

= attack angle of a plane hydrofoiL

= vertical distance from the still water level to the midpoint of chord length.

Jr, (n =1, 2, _{Bessel function of the nth order.} Y, L _{= force in the y-direction, i.e. lift.}

X, D _{= force in the x-direction, i.e.}

wave resistance.

= moment about the midpoint of chord length. = distance from the midpoint of chord length

to the center of pressure. = .drag-lift ratio.

= density.

=_- speed of advance.

C' _{= effective speed of advance.}

The subscript _{oo means the corresponding value in an} Infinite fluid.

TETSUO NISHIYAMA

INTRODUCTION

N A TANDEM type the after foil _{operates in the} wave from the forward foil. A possibility then arises for the elimination of a substantial portion of the wave effect by selecting a foil _{spacing so adjusted} to the design speed that the wave created by the forward foil is partially annulled by the wave from the after foil. Therefore a portion of the energy lost as wave resistance of the forward foil could be re-covered by the existence of the _{after foil.}

The object of this paper is to give a theoretical

explanation for the "wave _{resistance recovery."}

Regarding the waye from the forward foil as the stationary wave, which is a progressive wave in a steady state propagating with the same speed and in the same direction _{as the after foil, the} theoreti-cal expression for the lift, wave resistance and mo-ment is carried to the second _{order, and numerical} calculations are made. Curves are given showing the variation of the characteristics with the relative position to the phase of wave, and then the opti-mum position of the after foil_{is discussed.}

rDROFOILS NISHIYAMA GENERAL EXPRESSION

Consider the two dimensional motion due to a bmerged cylinder placed in stationary wave in-'ding circulation around the cylinder, the center

which is at a depth f below the mean water level.

ke the origin at the center of the cylinder,

th 0, horizontal and Or vertically upward. When e progressive waves propagate with velocity C in e positive direction of 0,, we add on the whole ass a uniform velocity of C in the negative di-ction of Ox; then the fluid motion becomes sta-mary, and the complex potential is

wo=Cz-I-Che-The boundary condition to be satisfied at the free l=x+iy; K0=g/C2.... (1)

w=Cz+Che1Kor+

F(K)e1x7c1K-0 F*(K)e-IK"-2KydK ... (4) We transform (4) in a form

ir

w=Cz+...Anz-n+.113nzn-F 277 log z where 11.

F(K)=.( i)nAn

_(n-1)! ( r

Bo=-Che-x01-'0 (nil)" K0" _{nil.') 1,,} F' (K)Kne-2xfdK Then, for the forces on the cylinder we have from

the Blasius' 1st formula

(5)

L

F(K)Fi« (K)K2e-'2KfdK

x=2,1,

_{im[Che-KF(K)K0}

[Che-Kf+4F(K0)K

30

zfr,

Y= 2=p Re _{F(K)F*(K)K2e-21raK]} (6)

For the moment about the origin we have from the Blasius' 2nd formula M = pi Im[CAI +Che-Ka-'0K0 dK0F(K)-;-F(K) :)" K±K,ikt F* (K) K KdK F(K) (K) le-2KfdK (7) surface is , Fw. aw aw

Re I --I-

p, 1 =0 at z=x+if. .... (2) eez2 az az

Then the complex potential of the disturbance by the cylinder is given in a form

00

K Ko

K--1-K _{F*(K)e-iicz-ncrdic}

w1=-_ F(K)elHzdIC

(3)

where the limiting value ix-00 is to be taken and the asterisk denotes the conjugate complex quantity.

Hence we may obtain the complex potential of

the fluid motion by the cylinder in a stationary wave, that is

(4')

To complete the solution of the problem in any

ven case, we have to determine the function

(K) so that the boundary condition of zero

,rmal velocity is satisfied over the contour.

INTEGRAL EQUATION

We take the cylinder to be an ellipse, of semi-axes

From (4') and (8), we have

c/o

dwidi;=Cpe'" sinh CiKCh e-IK°'+ pe'"sinh e1"sinh F(K)Ke'KzdK

0

±ipeinsinh Ke-iii,-2Krdic

a' and b', with its major axis making the acute angle with the positive direction of O.

In terms of a complex variable Z:=:;:+i-q, we take

z=p eie cosh (8)

and the contour is given by

$.=g0; p cosh =a'; p sinh

Now the 3rd term in (9) _{is written in a suitable} form in the elliptic coordinate

,T _ir

ipei"sinh Jo F(K) K KzdK= ₍₁₀₎

From (10), we get

iF(K)K=-2---J(Kpe19-1-iY13J(Kpe) (11) After some reduction, (9) may be reduced to

dw d=:.(Ce"td-D,1e-")+ ₍₁₂₎

where

C1=Chei13-Ko1 (i)nnJ,(K,,Pe")

(-0'n r

D.= Ch ei0-Ej (i)nnJ(Kpe1")

_{PIN+ (-01'}

_K-FK0im11

KK0itt

for n=1, C, has the additional _{term 1/2Cpe" and Di}

the additional term 1/2Cp-i°.

The boundary condition on the contour gives (13) Hence from (12) and (13) _{we obtain}

b --Chei13-Kof (i)nnJ(Kpe ")-L

_(i)"n

K-f-Kip

j0 K F*(K)J (Kpel°)e--21dK

c ; 1{-i-Ki, .

e'''S"

[

_{Che--113-KonJr, (Kpe-")n 1 -- ---'F(K)J(Kpei°)e-'"'clK}

K±ir.

(14)

with a similar expression for 111, including an addi-tional term,

1/2iCpe'°-1/ZiCpee-14

Putting (14) in (11), we havean integral equation

iF(K) K=rJo(Kpe")Y2Cp(e"e2.6.'e-")J,(Kpe")-F _{iChe-Kor(ei0H1+e-'5H2)}

2z-rx)

F* (v)G,e-2"dv+i z v+K-Fip.F(v)G,e-'="dv ₍₁₅₎

vK0-1,u

0 vK±ip.

where H1= 5'.(-1)nnJ,,(Kpe'")J(Kpel) H,. I:enfonJn(Kope-ie)J(Kpe") G1= :.(-1)nnJ(vpele)J(Kpel°) G2= 5.'eneonJ(vpe-' 9)J (Kpei LIFT, WAVE RESISTANCE AND MOMENT

OF A PLANE HYDROFOIL

We consider the limiting case obtained by making zero. The elliptic cylinder reduces to a plane hy-drofoil of chord length 2p, attack angle 0, and with

f(K)K= ikJo(KPeie)JI(Kpe")+ hcosecOe-K1(e1PH1-Fe-0112)

/

f(v)Gie-2"dv (17)

0 vKip,

f*(v)G,e-2vidv+

0 vKo+itt

where

Hi= V-1)"nJ(Kpe")J(Kpel)

H2= . :,nJ1(Kpe-i")Jn(Kpe1") F*(K)J(Kpei')e-2KfdK

its midpoint at distance f from the still water level. We write

F=271c.Cp sin61; F(K) =Cp sin0 f(0) ... (16)

then (15) is

G1=5;(-1)"nJ(vpei°)Jn(Kpel°)

G2= 5.ran(VPe-le)Jn(CPele)

A.S.N.E. Journal. February 1951 ₁₃₇

NISHIYAMA

HYDROFOILS

This integral equation is of Fredholm's type of the Developing the Bessel function in (17), it is suffi-2nd kind and may be solved by successive substi- cient, for the present purpose, to take as the 1st

tutions. approximation,

1 1

f (K) K= ik1/2Kpe"±ixiikK2p2e2 i 0 ± ._____1(3n3e31064ikK4p4e4"+ -16

'

1

-

cosecOe-Kor 1/2Kpe-10 -1/2Kpel"± 16-K3p3e0 -1/4K02p2e-2to x 1/8K2p2,210 +-1.-Kp3e-3"X 1/2KPel°±

-16

+e't3 {1/2Kpe" (-1/2Kpel°+-1-6-1K1p3e39)+1/4K02p2e2" x 1/sK2p2e21°

1

+-16K:133e"1° x1/2Kpel 6+

-Further we can develop G1, G2, H, and H2 in a similar manner.

Putting (18) into (17) and using the relation n!

r K+ Ko

Kne-2KrdK=K"-1 (r. - is) : r= _(2K000.1

+2{(n-1)! 1

(2Kf) n -± 2Kf (e2K"f) : s=2=e-2Kor

we obtain as the 2nd approximation

1

16

f(K).K

KID"e."" 64ikK'ple""134+

-= -ik-

V2KPei°131+1/4ikK2p2e'l(132-1---cosecOe-xof (1/8Kop2e2 °Kpr ,e' ± 1/8Kp1Kpr,e-113+ - - -) (19)

B, 1 -kKop(rsin0 + secs()) - 1/2Kp-r1+ 1/2kK"p3(r2sin30 +s cos0) +1/4kK11p" (r2sin0-scos9)

B,= 1 +1/4K02p2 (ricos20 ssin26) + 133= 1 1(Kp(rosin0 +scos0) +

B4=1

-Now we determine the circulation from Kutta's condition; after putting $,-=0 for the plane hydro-foil, we have

ir

277±.

-1)n(C-C,,*)=0

hence, we have (18) where

k=1Re

where

he-Ko'cosecBe -'0-1°A*K,p-pe r K±Ko-Fip,-1° f (K)KAi*e-2KrdK

0 K-Ko-l-ila

Ao=J,(K(Pe")+0,(Kapete) A,=J0(Kpele)+iJ,(Kpe'°)

..(20)

Carrying out the integration, (20) may be reduced

to

k=1-F a ,Kp+ aK2p2 -FaKO"p" +a4K04p4+ - - - (21) where

a = (rsin +scos0)

--pe-KofcosecOscos(0 + )3) a,= (rsine+scos0) + 1/2(ssin20r1cos20)

+1/2he-KocosecOesin(29+p)

a1= a, (rsin0+scos0) + (ssin20 rlcos20) + 1/2r, (rosin° + scos0)

+ 1/2scos0+ 1/4 (r,sin30+ scos30) +

1/4pe-"COS0 cos (30 +13)

a,

a4= a ,(r0sin9-F scos0) + _{(ssin20r1cos20) + -yr1(rsine +scos0)}

2

+ 1/2a iscos9 1/4_{(r2sin0+scos0) (rsin0+scos0) + (r3cos20+ssin20)}

16 +1/8r+1/8 (rcos20ssin20) +-- (r,cos40ssin40) 16

lb

. e Ku' cosecOeson (40 + /3) 16 p

Using (19) in (5) and (6), we have

L/L =kKpsinO[iCr+kKpr,sinO-F-K'pl 1/2k2s sin29

k2sin0sr, (rsin0+s cos) 1/4r, } +1(03p 1/2kr,(rsin0+s cosO)

1/2krsin0-1/4kr, (1/2sin39sin0) }-+ -

-+ e11...P[k+ 1/2K,,p sin9 1/2K02p2{ k sin0 (rsin0+ s cos0) +1/2k cos 20 } 1/4Kp3 (risin0+ 1/4sin30) + - - - ]sin/3

h

e-",".Kp[1/2Kp cos0 1/2K0P21 k cos0(rsin0 +s cos()) + 1/2k sin20 1/4K03P3(r,cos0-1/4c0530) + - - - ]cos/3 (22)

D/L, =Ka) sin() [k2s+ kKps sin0 K02132 1/2k2r,sin20

k2sin0es(rsin0+s cos0 ) 1/4s } +1CO2133 1/2ks (rsin9+s cosO) %kris sin() 1/41(s(1/2sin30sinB)

- - - ]

he-Kolcp[k-F1/2Kopsin0-1/21(02p2i k sin 9 (rsinB-Hs cosb) + 1/2k cos20 1/41(0"p" (risinG-1/4sin30) - - - ]cos/3

he-"Kp[1/2Kopcos0 1/2Ko2p2.{ k cos& (r)sin0+s cos0) +1/2k sin20

1/4Kep"(r,cos8 1/4cos30) - - - ]sin/3 (23)

A.S.N.E. Journal February 1959 ₁₃₉

And also, from (7) and (19) we have

M =1 Kp tanO[k rcos0s sine +(rsin0 +s cosO)cotO

}-Kp; 1r,cot0+ 1/4s k2( rcos0 s sine) (rsin0 +s cos0) I2k:(r1sin20-1-s cos20)

-h

+ - e koKp sece[cose+kKp sin20 (r0sin0+s cosO)cos0 - - - ]sin/3

h__{e-"oKp sece[sin0klcp cos20 +(rsin0+s s cos0)sin0} - - - ]cosfl.. (24)

Taking a special case h=0 in (22), (23) and

(24), we obtain the corresponding expressions of he submerged plane hydrofoil in a uniform stream.

Hence for the center of pressure we have /M

h/h =--

_{L 'L,,}

(25) Now as a rough guidance for the variation of per-formance of a plane hydrofoil among waves, it is useful to estimate approximately the variation of drag-lift ratio, that is

(

f

where E, may be obtained by referring the experi-mental results in the wind tunnel.

NUMERICAL EXAMPLES

We take the following case as a numerical exam-ple:

attack angle 0=5°, immersion t/f=1/2 (t = 2p) amplitude of stationary wave 2h/t=0.2

phase of stationary wave /3=0, (max. wave slope) /3=7/2, (zero wave slope) The nature of stationary wave taken in this

ex-ample is given in Table I.

TABLE I

The variation of lift may be attributed to the variation of the effective velocity at the positionof the hydrofoil and also the circulation, i.e. the effec-tive attack angle; generally we have

xC'/C

Then in order to know the relative importance, the variation of lift and circulation is shown in Figure 1.; from which we can see that the greater part of (26) I.0 05

L.

9 direction, of advance

=j1T

If ff 2 3 Figure 1. Lift.

the variation of lift is due to that of circulation or effective attack angle and the variation of effective velocity is very small. The former is mainly due to the variation of the direction of streamlines by wave-making among waves, and the latter increases under the crest and decreases under the hollow

only by the orbital velocity of the wave.

When the phase of waves just over the hydrofoil is a crest (pr----7r/2) and midpoint from hollow to crest (/3=7), or hollow (fl=3.7r/2) and midpoint from crest to hollow (3=0), the lift is respectively

larger or smaller than that in a uniform stream. However the variation of lift is considerably larger in the maximum wave slope than zero wave slope; the reason .for which can be interpreted to lie in the variation of effective attack angle.

----in stationary no,

--

tt

still water 10, 20 40 60 30 100 120 Alt 5 20 40 80 120 2h/A 1/25 1/100 1/200 1/400 1/600 c/ Nigf- 0.63 1.26 1.78 2.52 3.10 IYDROFOILS NISHIYAMA

NISHIYAMA 002 0011-2 p=o -D

//

40 60 80 100 120 1 1 1 1 1

---

Lit Still water

2 ₃

6/rff

Figure 2. Wave Resistance.

Curves are given showing the variation of wave

resistance due to phase of wave in Figure 2; when the phase fl is 7r/2 and 0, or 377/2 and r, the wave

resistance is respectively larger or smaller than

that in a uniform stream; this _{comes from}

strength-ening or annulling between the stationary wave

and the wave created by the submerged hydrofoil. Hence the wave resistance recovery is maximum when the phase of the wave just over the hydrofoil is a mid-point from hollow to crest.

in still ureter

A

T

o io 20 40 60 80 100,120 0 I

CA 2

3

Figure 3. Drag-Lift Ratio.

1.0 05 05 9 0 Figure 4. Moment.

Curves are given showing the performance of the hydrofoil among waves by obtaining E, from the experimental results in Figure 3. The performance deteriorates under the phase 13=37/2, in particular too much under the phase 13=0, and on the contrary improves under the phase /3=7/2, in particular too much under the phase 13=7.

Curves are given showing the variation of

mo-ment in Figure 4. The momo-ment is larger under the

phase )3=7/2 and 0 or smaller under the phase /3=37/2 and 7 than that in a uniform stream. Hence the center of pressure moves to the leading edge

A

T

II

I I 101 10 40 60 2 ₃

-C/RT

Figure 5, Center of Pressure.

HYDROFOILS

--

in still water N

T

, 1 1 101 20 40 60 80 100 120 ! 810 100 120

A.S.N.E. Journal, February 1959 ₁₄₁

TT

... ,.,... , ..,

HYDROFOILS

under the phase 13 =37-/2 and 0, and on the contrary to the trailing edge under the phase )3=r/2 and r,

from a quarter point of chord length, as shown in Figure 5.

CONCLUSION

The expressions of the characteristics of the sub-merged plane hydrofoil among stationary waves are obtained and, in particular, the variation of the characteristics with the relative position to the

phase of waves is clarified by the detailed

numer-ical calculations.

When the phase of wave just over the hydrofoil is a mid-point from hollow to crest, the lift and the

This paper of Professor Nishiyama, the third of the series published in this JOURNAL, is a logical

and important sequel to the previous studies. The

problem of designing the desired lift and drag characteristics into a hydrofoil craft is an extremely complex one. This paper goes far along the path of easing the burden of design, and yet represents only a part of the theory and experiments to be devel-oped.

The effects of separation of the foils is an emi-nently practical problem. The need for a direction-ally stable hydrofoil craft leads directly to a system involving two or more tandem foils.

It is an excellent principle of engineering to try to recover energy which ordinarily might be dissi-pated. As an example, the naval architect places his propellers at the stern of his ship in order to regain, in the wake, energy transferred to the water as re-sistance of the ship. In the case of the tandem-foil hydrofoil craft, energy is transferred to the water by the forward foil, in terms of viscous resistance, vortex shedding and in the creation of a wave sys-tem that travels aft of the foil. For most practical craft in a real fluid, the large separation of foils (in the order of 15-25 chord lengths) results in the fact that the effects of viscous resistance and vortex

shedding ("downwash") have essentially

disap-peared. The wave train, that the aft foil responds to, is the major physical phenomenon that offers the possibility of recoverable energy.

In light of these remarks, this paper of Professor

DISCUSSION

"Hydrodynamical Investigations on the Submerged

Hydrofoil"

Part III by Tetsuo Nishiyama

*By Dr. B. Silverstein, Ship Design Division Bureau of Ships

NISHIYAMA

wave resistance are respectively larger or smaller

than that in a uniform stream. Hence we can safely conclude that the after foil should be arranged un-der the midpoint from hollow to crest of the waves created from the forward foil.

REFERENCES

Milne-Thomson, "Theoretical Hydrodynamics," 2nd Edi-tion. p. 365.

T. Nishiyama, "Hydrodynamic Investigations on the Sub-merged Hydrofoil. Part I." JOURNAL OF THE AMERICAN

SO-CIETY OF NAVAL ENGINEERS, Aug. 1957.

Flachsbart, "Messungen an ebenen und gewolbten Platte," IV. Lieferung, Erg. A. V. A. Zu Gottingen.

Nishiyama's becomes of great value. Here we are given guides as to the optimum separation of the foils so as to maximize lift and minimize drag.

It is stated in Professor Nishiyama's develop-ment that he derives the forces on a foil in the

presence of a wave train, presumably created by some forward foil. This wave train moves with the same speed as the foil so the wave system appears as a stationary one to the foil. This is a special case

of a situation which is of much significance in the

design of hydrofoil craft, the stability in waves.

This special case is identical to the one where the

craft is operating in a following sea traveling at the same speed as the craft (the period of encounter is infinity). An extremely useful extension of the theory developed in this paper would be the

calcu-lation of forces in non-stationary waves. In this con-nection, it might be mentioned that a number of tests have been run of various tandem foil arrange-ments. In DTMB Report 1140, Leehey and Steele present some experimental and theoretical results of the motions of hydrofoil crafts with tandem vee-foils, vee-foil forward and flat foil aft, and vee-foils with a flat mid-foil. Continuations of this study are soon to begin at the University oi'Minnesota. In addition, a soon-to-be published report by F. Ogil-vie of DTMB will present a theory of the stability of hydrofoil craft (consisting of tandem foils) in waves. Dr. Paul Kaplan, in an also

soon-to-be-pub-Although we failed to include a credit line, the discussion on Part II in the November 1958 Issue. was also written by Dr.

NISHIYAMA _HYDROFOILS

lishecl report by the Experimental Towing Tank, indicates that Ogilvie's theory should be modified, in order to take into account the effect of the waves, created by the forward foil, on the aft foil. These studies, dictated by the important practical problem of stability, indicate another avenue of needed theoretical work. I would strongly suggest that Pro-fessor Nishiyaxna, or some other investigator having the theoretical bent, investigate the forces devel-oped by a vee-foil; that is, a surface-piercing foil. The sequence of problems attacked by Professor Nishiyama is appropriate: 1) the determination of the hydrodynamic characteristics of the

surface-piercing foil, 2) the inverse

problemdetermina-tion of the profile from a prescribed pressure dis-tribution and 3) the determination of the hydrody-namic characteristics of the after foil as affected by the waves created by the forward foil. As step 4),

as suggested earlier, the forces and moments of a foil operating in .a progressive wave system would be of great value.

In a discussion of a theoretical paper like this one, it is useful to condense the author's procedure, along with various comments.

Prof. Nishiyama initially considers a cylinder in a stationary wave train (zero relative speed). The long axis of the cylinder is perpendicular to the di-rection of motion. There is a circulation, I', around the cylinder (a lift force is developed). The two-dimensional velocity potential is developed, satisfy-ing the free-surface condition (equation (2) ) the free surface is a streamline. The complex velocity potential is then expressed in terms of the unknown function, F (K), and the circulation, i'. Using the Blasius' formulas, given in complete form in my

discussion of Prof. Nishiyama's first paper,

pub-lished in the August 1958 JOURNAL, the expressions

for lift and moment are given by equations (5), (6) and (7). At this point, a special cylinder is con-sidered, that is, one whose cross-section (taken at an angle e to the direction of motion) is an ellipse. Elliptical coordinates are used to express the equa-tion of the cylinder and the derivative of the ve-locity potential (which yields the veve-locity in the direction of the derivative). The normal velocity on the perimeter of the elliptic cross-section must be zero. This condition (equation 13), sets up an in-tegral equation in r and F (K).

As an approximation to a hydrofoil, the minor axis is set equal to zero; the result is a

two-dimen-sional flat plate, of chord equal to 2p, and at an

angle of attack, i). The result is then a somewhat 5implified form of the previous integral equation. A second order approximation to the solution of this equation is obtained by an iteration process.

The Kutta condition, the velocity at the trailing

dge is zero, then yields a solution for the

circula-ion. The now determined form of the function P(K) is inserted into the Blasius' equations for

'orce and moment and the result is equations (22), (23), and (24) for lift, drag and moment of the flat

plate at an angle of attack in a stationary wave train.

It should be noted that the forces and moments are non-aimensionalized by equivalent forces and moments in an infinite fluid (no free surface), with the exception of drag, which is non-dimensionalized by the lift force. This served to emphasize one of

the basic assumptions of this development; this

theory assumes a perfect fluid (zero viscosity). In the case of a two-dimensional foil at an angle of

attack in an inviscid infinite fluid, there is a lift

force, but no drag (D'Alembert's paradox). DID°, would be equal to as D =0. D itself is not equal to zero, as even in an inviscid fluid, the presence of the free surface causes a drag force due to the cre-ation of waves that are carried to infinity. The con-cept of symmetry aids in resolving, intuitively at least, this paradox. In the infinite fluid, the condi-tions far ahead and far astern of the two-dimen-sional foil are identical; no energy has been lost. However, in the case involving the free surface, there are no waves far ahead but there are waves far astern; this asymmetry shows that energy has been transferred from the foil to the water.

The numerical examples developed by the author are extremely interesting. When the center of the foil is midway between the crest and the trough of the waves, the lift is at a maximum and is greater than the case of the foil traveling in undisturbed

water (see Figure 1). It may also be noted that

Professor Nishiyama has also plotted the circula-tion. This clarifies the picture in that it shows that the major effect on lift is the change in circulation and not the orbital velocity of the wave. In Figure 2, we see that the drag, in this optimum position for lift, is actually reduced. At some low values of the Froude number based on submergence

(C/V gf), there exists a thrust on the foil instead of a drag. This appears truly to be almost "some-thing for no"some-thing."

The more exciting result, from the designer's

point of view, is the wedding of Figures 1 and 2 into Figure 3, the drag-lift ratio. The inverse of this ratio, lift/drag, is one of the usual criteria of good design of a hydrofoil craft. A great deal of care is exercised to maximize lift/drag. Figure 3 shows

clearly (and this has been borne out by

experi-ment) that considerable improvements in lift/drag are possible by a proper selection of foil separation. It is worth mentioning that Professor Nishiyama has considered the foil acting in a wave train. The

properties of these waves, as related to the foil causing them, is not presented. The wave properties, however, are readily developable from extensions of Professor Nishiyama's paper in the August 1958 JOURNAL, or from other theories of the flat plate

hydrofoil.

Unfortu. nately, all the experimental data I have seen have been classified, as a security measure. These data, however, are derived from, unsystematic