Lifting line theory of the submerged hydrofoil of finite span

SYMBOLS C=. speed of advance f = immersion t= chord length b=span A= aspect ratio Ca= lift _coefficient Cw= resistance coefficient

W = coefficient _{showing the influence}

of viscosity on

the lift r2= circulation

y= non-dimensional _circulation

['he suffix oo _{means the corresponding value in an infinite}

Kiid and the dash denotes the_{corresponding value in the} wo dimensional _case.

INTRODUCTION

From the viewpoint of the direct _{application to} ractical problems, the

_{theory of the}

_submerged ydrofoil of finite _{span is of extreme}

importance. rp-to-date some theoretical _{treatments of the} sub-zerged hydrofoil

_{as a three dimensional}

hydro-ynamical problem have been published. These re-'arches are worked _{out by Weinig,"- Mauro' and} 'u.3 However, of these works, Wu's _treatment'

ems to be the most prominent since both_boundary nditions, on the free surface and on the _{contour of} e hydrofoil, are taken into account _precisely.

TETSUO NISHIYAMA

LIFTING-LI1VE

_THEORY

_{OF THE}

SUBMERGED

HYDROFOIL OF FIIVITE

_SPAN

THE AUTHOR

studied Naval _{Architecture 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 connected with general

hydromechanical problems in Naval Architecture, in particular with wave resistance theory _{and with the submerged}

hydrofoil. Professor Nishiyama

was the author of the three-part paper titled "Hydro-dynamical Investigation

on the Submerged Hydrofoil," which appeared in the August 1958, November 1958, and_{February 1959 editions}

of THE JOURNAL.

PART

n-A calculating method for obtaining _{hydrodynamical}

characteristics of the submerged hydrofoil of finite span.

Nevertheless it seems that improvement still re-mains to be made in the following areas:

The expression of velocity potential is too com-plicated to understand the physical _{meaning of} the equation used in the calculation.

The methodfor _{obtaining the characteristics}

is too troublesome to adopt from a practical _point of view.

,

a The flow _{around the hydrofoil can not be justly} represented by arranging the bound vortex to the trailing edge.

In this paper the author, in the direction _of im-proving the above-stated _{weak points, has developed} the lifting-line theory of the submerged _hydrofoil of finite _{span and examined in detail}

the following items:

Part I. A practical

calculating method for obtain-ing hydrodynamical _{characteristics of}

the

sub-merged hydrofoil of finite span.

Part II. A practical

_{calculating method for}

ob-taining hydrodynamical _{characteristics of the}

sub-merged hydrofoil of dihedral angle.

Part la A

_{practical calculating}

method for de-termining the distribution of chord length and attack angle from the_{optimum distribution of}

circulation of the submerged hydrofoil.

Part, IV. Side wall _{effect of an}

experimental tank

A.S.N.E. Journal, August 1959

ARCH1EF

_Lab.

v. Scheepsbouwkomdf.

Technische

_Hogeschool

SUBMERGED HYDROFOILS _NISHIYAMA

on the characteristics of the submerged hydrofoil. Part I

will be covered in this issue of The

JOURNAL. The remaining parts will be included in subsequent issues.

VELOCITY POTENTIAL

We take the origin of coordinate 0 on the still

water level with ox-axis in the direction of advance,

oy-axis in the direction of the span and oz-axis

vertically upward, as shown in Figure 1. Because

of a fairly large aspect ratio and small loading,

Prandtl's original concept of the lifting-line may b6 applied to the submerged hydrofoil of finite span; the hydrofoil can be substituted by a bound vortex, namely lifting-line, and the free vortex is situated

on the plane parallel to the xy plane. Figure 1. Coordinate.

When a hydrofoil of span b advances in an infinite fluid with the constant speed C at the distance f from the origin, the velocity potential is given by

10

b/2 _{a f} 1

.J

47 -07, -b/2 az l.-\/(x-4)2+

(y)2-1- (z+02

}dr/

Differentiating by z and then integrating by t, (1) can be transformed into the form

fb/2(IT

oo di

=

r(71)C17) secede e-K(z+r)4-igrvdK 87r2 -o/2 0 1 bj2 CO

-A

1'(77)&71 e-tilk+ocosu(y-Odu where rv=xcos0+(y-n)sin0

As the second term does not inc\ide the variable x, it may be interpreted that the first term represents the fluid motion by the bound vortex and the second term by the free vortex.

When we denote the velocity potential of the submerged hydrofoil 43, the boundary condition on the free surface can be given by

a243 a(1) ag,

+K.

0.

g/C2

ax-, " az ax '

Then the velocity potential can be expressed in the following form, from (1') and (2):

CP=4)00 +433.-1-cD2 (3)

b/2 7 co

cl,i= r(n)c1771 seced0 1 e-X(f-z)-"K7dK

87.2 _-7 0 1 i 1)/2 CO ' r(n)(1771 e-u(r-z)cosu(y-)du (4) 47r -b/2 o i rb/2 r a7 ico e-K(f+z)+11C7

4,2 =871.2K0 )..bi2 iI1(77)dn sec39d0 dK .

.- \ 0 KK0sec2e-FinsecO

(5)

\

where the real part

is

to be taken, and the Rayleigh's viscosity

co-efficient ILis to be set equal to zero after'Sserving its purpose.

The velocity potential 43, is the imag of it'co when the free surface is con-ceived as a fixed boundary; the first and second terms represent the fluid motion by the bound and free vortex respectively at the image point. The velocity potential 4,2 represents the wave disturbance.

512 A.s.N.E. Journal, August 1959

NISHIYAMA SUBMERGED HYDROFOILS

Now we transform 4)2 by applying the contour integration: For > 0 00 K f h/2 f msinm(fz)+Kosec20cosm(fz) 4)..=

r( )d

sec30e-"wilOdm.(6) _br2 M2-1-Ko2sec40 00 For 77<0 21(7.°2 Lbt:/22

r

_{) dr) 1} +°° jr oc° Ko 6/2

rl-vo

77 -6/2 msinm(fz)+Kosec20cosm(fz) sec30067d0dm m2A-K02sec40 sec30e-K.(f-2)sec2°cos(K07vsec20)d0 (7)

yn

where

(6) and the first term of (7) represent the local symmetrical disturbance

which can be observed only in the vicinity of the hydrofoil, and the second term of (7) represents the regular wave motion to the rearward of the hydro-foil.

We transform the second term of (7) by putting u equal to tan 0

1)/2 , -00t0o

[4)21wave= r(n)Cin (112+1)112e-K°"-z)(1'"2)cos[Ko(1-Fu2)1/2{x+ (yn)uHdu -b/2

(7') Generally the speed of advance of the submerged hydrofoil is extremely high, therefore it can be easily shown that the contribution to the integral (7') is exclusively due to the larger values of u. Then paying attention only to the vicinity of the hydrofoil, we obtain

lim [(1),Lave= 1'12 r(n)d.ri cosKo(y-4)u2}e-Ko"-z)"2udu

Ko-s0 -1/2

X --0

Transforming (7") by putting Ku2 equal to t, we have

urn 4/2 r(n)dnicOs t (yn) }-e-t(f)dt

(7")

110.0 Z7 -6/2 0

xo

Now comparing (7'") with (4), the following relation holds

Jim [42,2]ve= 2x [the 2nd term of (DJ (8)

Ko-,0

In other words the velocity potential of the regular wave at an extremely high speed is equal to twice that of the image free vortex; or the wave motion immediately after the hydrofoil gradually approaches the free vortex motion as the speed becomes higher.

On the other hand the corresponding relation 4 in a two dimensional case is

lirn [4321wave=0 f (9)

K 0

That is, the regular wave motion immediately after the two dimensional hydro-foil can not be observed at all.

The relations (8) and (9) will be useful for considering the relation between the induced and wave resistances and the variation of the wave resistance with the aspect ratio.

SUBMERGED HYDROFOILS

INDUCED VELOCITY

Now we consider in detail the induced velocity before examining the forces of the hydrofoil.

Vertical induced velocity

Differentiating (3) by z we obtain 1 r br2 dr(g) d 1 b/2 r()dnIccue-2urcesu(yn) 47 _b/2

dn y-7/

47 -1)12 0 K21 b/2 7/2 r(q)dn sec5Oe-2Kofec2°cos-/Kosec=0*(yn)sin0 k161 (10) 7o

Evaluating the integral after altering the variable of the integration, (10) can be reduced to .c b12 dr(g) di? 1 f b/2

r()k(yn)dn

(10r) ' 4r -12

Cin y

- -bi2 where 4f2y= Ko= 4f=

k(y) e-}cor [(I+ )K0

4f2+372 2 3,24-4f2

+2 2f 1 8f2

3'2+4f2 KoV Y2+412 +K(y=-1-4f (11)2)3/=

with the modified Bessel functions of the second kind Ko and R1, the argument of which is --(y2+4f2)1/2.Ko

2

(11) can be transformed for extremely low and high speeds as 4f2 y2

limK(y)=limK(y)=

_(y2±4f2)2 (11')

Ko-.7 Ko,,

Therefore the vertical induced velocity is caused by the regular wave Motion and free vortex motion, and further, decreases at extremely low speeds and on the contrary increases at extremely high speeds. Correspondingly to this, the effective angle of attack increases at extremely low speeds and decreases at extremely high speeds.

Horizontal induced velocity

As easily shown from (3), the horizontal induced velocity has no relation

at all with the free vortex. Therefore if the aspect ratio is very large, the

horizontal induced velocity could be replaced by that of the two dimensional case.

The horizontal induced velocity in two dimensions 4 is given by

r f

w'

_x 2Ko e-K(f-311-iKxdK (12)

27 J 0 271. J KKo+iii,

Applying the Fourier's double integral theorem to (12) we obtain

COD oo

w.'= l''(71)(177 cost3(yri)dg e-0(1-3)+10xda

1Go

yr e-0(f-y)-1-i0x .

----2K0S.T(n)dn cos13(yn)dp . a . (12')d

2r2 o o aKo-F1P,

Then putting a =Kcos0 and /3=Ksin9, (12') can be transformed into 1

8,2[r(y)di .c it dB cipKe-K('-.)"80I1 xc05o+(Y-70sine .dK

-7r 0 _ _{oo Ke-K(f--z)c050+11: xcos0+(y-T7)sIn0} +2K0ir(7)dn secOdO g _{dK (12")} 0 KK0secOd-ipsect9 NISHIYAMA ,

---T

NISHIYAMA

SUBMERGED HYDROFOILS As it can be easily demonstrated that the contribution to the integral _with

respect to 0 is mainly due to the

extremely small value of 9 if the aspect ratio is very large, (12") is reduced to

1

--[ f r(77)d77 j. 7 dB Ke-K(f-z).+iK xcose+ (y-71)81 no ).di< 87-2

-7r o

w j-oc _{Ke-li(t-z)+11: I:case+(y-V) s}

I no

+2K jr(n)dn I secgde9

dK _(12'")

-7r

0 KKosec2G+iisecO On the other hand, _{by directly differentiating}

(3) with respect to x, (12" ') can also be obtained. Therefore_{the relation holds:}

n 1

w,,

r()

. _{{--Koe-2Ko1Li(e21f )}}

(13)

7r 4f

where Li is a Logarithmic integral.

In short we can see that the horizontal _{induced velocity depends}

only on the local symmetrical disturbance and the bound _{vortex motion, and is negligibly} small compared with the advancing speed of the hydrofoil.

INTEGRAL EQUATION The circulation in two dimensions is given by

m

tr=

-

_{Ca; m2rw}

(14)

2

Then the circulation on the span of the submerged_{hydrofoil is given by} r(y)=- _{t(y){Cwx(Y) 1(4}

a(37' _Cwx(Y)w,(Y) (15)

On the other hand the_{circulation in an infinite fluid is given by}

r,, (y)=y9in t(y)C a(y)

(16)

Taking a fairly large aspect ratio and small loading _{into account, and} neglect-ing the 2nd order, we have from (10') , (15) and (16),

b/2

r(y)= t(y)roo.fc-(y)cb7+

_{rap (3)}

(17)

8r -b/2

When the circulation in an infinite fluid is given, the _{corresponding value in} the submerged condition can be known by solving this _{integral equation.}

From the circulation, the lift and resistance can be obtained respectively _by

L=pC1b/2_1)/2r(n)dn ₍₁₈₎

b/2

D=pf r(n)w,(n)c171

(19) NUMERICAL SOLUTION

An analytical solution for this integral equation (17) is too troublesome _to obtain. However from the _{fact that (17) is of Fredholm's type of the}

second kind and the kernal K -(y-77) _{is regular, we can solve numerically by}

applying Nystrom's method.

Putting

(17) is reduced tor(y)=2bCy(y) and 1 (y) =2bC (y)

t() r

YtY)

b _{yfn)KfY-77)d9+7}

(Y);IY!<1 (17')

-1

SUBMERGED HYDROFOILS where

R o

482[32 1/4P'1372 .. t- - -KO 482 \

K--(y)=-- 1 -,;., v 6-.7-..t.., +2(Kof)- ( 8 ) e [{ 1+ ., ..

.., }X K(

.) .. V 48-+IAP y2)

4o- + bip-Y- t- 48-+

1/418-y-with

III

t( y)

y(Y;)= _87-b0 tkY;) 'yai"-Y(ni)it(37Jni)+yx(Yi)

Taking seven points over the span after Glauert's method, seven simultaneous equations are obtained,

but we can reduce them to four simultaneous

equations owing to the symmetry of the distribution of circulation with respect to the central plane. Then the coordinates 7/i and coefficients a," in the

nu-merical calculation should be taken as follows ( = 0.06474

(a27)=0.13985 )=0.19091 a47 =0.20898

Thus using the numerical value of y (y), the lift and resistance can be also obtained from (18) and (19) by a similar numerical procedure.

NUMERICAL EXAMPLES

Numerical calculations are carried out for the fol-lowing cases:

Aspect ratio X=5 and 7

Ca,

7,0(0= 7rAV 1-77 2 )= -±0.94910 777 77' )= ±-0.74153 776 773 )=-±0.40584 n5 714 =0 Circulation Immersion f/ty=0=1 Coefficient w=0.85

The distribution of circulation over the span is shown in Figure 2. Except at extremely low speeds, we find the effective circulation distribution only small compared with that in an infinite fluid for the speed range of interest. The variation of the lift and resistance with the Froude number are shown for three aspect ratios X=5, 7 and CO in Figures 3 and 4

respectively; from which we see that, compared with the corresponding value in the two dimensional case, the effect of the free surface on the lift and wave resistance decreases as the aspect ratio becomes

+2{ 28

482+ y4p2y2

+ K,,f{482+1/4fl2y2}3/2}V 48-Kof ,+1/0-y- )1

8=fity_0 and

Then by Nystrom's method (17') is transformed into

1 8 i?V 482+1/4[32Y2 004 002 0 -1.0 0 LO

Figure 2. Effective Distribution of Circulation for the As-pect Ratio 5 (Dotted line shows the corresponding value in an infinite fluid.) 1.2 1.0 0.6 (17") /3 =bity,.. (1r") NISHIYAMA 0 1.0 20 3.0 40 5,0

Figure 3. Variation of Lift with the Froude Number for the Aspect Ratios 5, 7 and co. (Dotted line shows the cor-responding value in an infinite fluid.)

smaller, and also the wave resistance has a limiting value at extremely high speeds. The reason for the latter may be explained by the relation (8).

In order to confirin the accuracy of these results

for n=7, we compare the corresponding results

for n=9. The distribution of circulation is shown in Figure 5 and the lift is as follows:

r(0)/2bC Ca co Ca /Cnco

n=7 0.03869 0.8780

NISHIYAMA 006 0.04 002 0

rfr

0 1.0 2.0 3.0 4.0 5.0

Figure 4. Variation of Resistance with the Froude Number for the Aspect Ratios 5, 7 and Infinity. (Dotted line shows the corresponding value in an infinite fluid.)

2bCC0.03 0.04 _t o o aspect ratio 10 4 -1.0 0 10

Figure 5. Effective Distribution of Circulation for K f =0.1 ind A=7.

From these we see that the results for n=7

are iufficiently satisfying for practical purposes.

COMPARISON

In order to compare with the experimental

re-mits 5 published recently by NACA, we arrange

iur calculations as follows:

The replaced lifting-line is reckoned as a hori-:ontal bound vortex through the quarter-chord point rom the leading edge, because the center of pressure n the ordinary hydrofoil section almost coincides vith this point.

The coefficient showing the influence of viscosity

akes a value

between 0.80 and 0.90; therefore ye select 0.85 as a mean value.

Comparisons are made for the two conditions as

ollows:

immersion

0.84 0.59

'he lift and resistance are shown in Figures 6 and 7, nd Figures 8 and 9, respectively. Surveying these

1.0

05

SUBMERGED HYDROFOILS

NACA TANK NO.1 NACA TANK NO.2

4 6 10

Figure 6. Comparison of Lift with NACA Data for the As-. ect Ratio 10.

A NAGA TANK NO.1

0 1 4 6 8 10

Figure 7. Comparison of Resistance with NACA Data for the Aspect Ratio 10.

1.0 0.5 Figure 8. Comparison pect Ratio 4. r_ Cur c2 vat:,

0 NAGA TANK NO.2

4 6 8 10

of Lift with NACA Data for the

As-°NAGA TANK NO.2

0

I

0 2 4 6 8 10

Figure 9. Comparison of Resistance with NACA Data for the Aspect Ratio 4.

Cu, A = 5 ace 008 0.2 0.1 002

SUBMERGED HYDROFOILS

results, the calculated values are in good accordance with the experiment for an aspect ratio of 10 but larger than the experiment for an aspect ratio of 4. The cause of the latter may be interpreted as the effect of the lifting-surface. At any rate, taking into account that the aspect ratio of the submerged hydrofoil is relatively large and almost 9 as a mean, it can be safely said that this present method gives a satis-fying result for the normal condition of operation.

CONCLUSIONS

In this paper the lifting-line theory of the sub-merged hydrofoil of finite span is developed. The main problems treated are as follows.

Introducing Rayleigh's viscosity coefficient, the velocity potential is represented in a relatively simple form and its physical mean is illustrated.

A practical calculating method for obtaining the characteristics is proposed by applying Nystrom's method to the integral equation of circulation.

By the numerical examples the effect of the as-pect ratio on the lift and resistance is clarified. Also,

In this paper, Professor Nishiyama studies an im-portant practical problem in connection with hydro-foilsthe finite wing. As is pointed out, Wu's treat-ment of this problem is elegant but is not in the form that is easily used by the designer; there are a num-ber of integrals that are intractable.

Before examining Professor Nishiyama's work, it is useful to summarize the background of the lift-ing-line theory, due to Prandtl, which both Wu and Nishiyama have extended to consider free surface

effects.

It should be re-emphasized that all of the studies of lifting surfaces (wings, hydrofoils, etc.) are based on the vortex, and systems of vortices. These vortices can be seen in flow pictures of lifting wings, but for our purposes they can be considered as a special sort of mathematical singularity. It may be consid-ered as a point that induces a special velocity field about it. All particles travel in a circular path about the vortex with a tangential velocity that is inversely

proportional to the distance from the vortex (as

the path is circular there is, of course, no radial

velocity). There is a subtlety here that might be

noted; the particular relation of tangential velocity to distance from the vortex is a result of insuring that the motion is "irrotational," where the laws and theorems of hydrodynamics are well-known. The

DISCUSSION

"Lifting-Line Theory of the Submerged Hydrofoil of Finite Span" PART I

by Tetsuo Nishiyama

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

NISHIYAMA

comparing the theoretical value with the experi-mental, the reliability of the proposed method is confirmed.

The author wishes to express his gratitude to

Prof. F. Numachi and Prof S. Fuchizawa for their many inspiring discussions.

REFERENCES

F. Weinig, "Zur Theori des Unterwassertragflugels und

Gleitflache," Luftfahrtforschung 1937 Bd. 14.

H. Maruo, "Effect of the Water Surface on the Submerged

Wing," Jour. of Nay. Arch. of Japan, vol. 86, 1953.

Y. T. Wu, "A Theory for Hydrofoils of Finite Span," Jour.

of Math. and Phys. vol. XXXIII, No. 3, 1954.

T. Nishiyama, "Hydrodynamical Investigation on the Sub-merged Hydrofoil," JOURNAL OF THE AMERICAN SOCIETY OF

NAVAL. ENGINEERS, Aug. 1958.

K. L. Wadlin, "A Theoretical and Experimental Investiga-tion of the Lift and Drag Characteristics of Hydrofoils at Subcritical and Supercritical Speed," NACA Rep. No. 1232,

1955.

G. Higgins, "The Prediction on Aerofoil Characteristics,"

NACA Rep. No. 312, 1929.

particles, though moving in a circular path, do not "rotate"; they stay in position very much like the cars on a ferris wheel.

If we construct any closed curve surrounding a vortex and integrate the tangential velocity from any point on the curve, traversing this curve back to the same point, we derive the circulation, I', which has

units of feet squared per second.

The laws governing vortices have been studied by many men including Helmholtz, Lord Kelvin and Kutta. The latter developed the famous theorem of lift which sparked the great interest in vortices and their associated circulations:

Lift per unit of span = Mass density x velocity X circulation,

or _L=pVr

Any explanation, of vortex representations for lifting wings must satisfy the laws of vortex be-havior, such as circulation is a constant with respect to time, vortex filaments must close, etc.

Now let us consider a hydrofoil at an angle of attack, moving with constant velocity through an ideal fluid. Immediately, a so-called starting vortex (parallel to the span) leaves the wing. The strength of this vortex is equal and opposite in sign to the

NISHIYAMA _{SUBMERGED HYDROFOILS}

bound vortex remaining on the foil: This is con-venient, as one of the laws to be satisfied is that the circulation is constant with time. Both vortices (the starting and the bound) extend to infinity and thus satisfy another law that vortex filaments must close. The finite wing, in order to satisfy this latter law, must shed vortices at each wing tip; this is called the trailing vortex. When the starting vortex is far downstream the system consisting of a bound vortex on the foil and the trailing vortex form a representa-tion that looks somewhat like a horseshoe, and thus is commonly referred to as a horseshoe vortex field.

This leads to a fundamental difference between

the two-dimensional and the finite foil. For the

former, 1 _{is a constant with respect to the span. For} the latter, the circulation must vanish at the foil tips and thus must vary with spannote, for example, in equation (1) of this paper that

r

is a function of 7i as it varies from one tip to the other.

The horseshoe vortex system no longer preserves the symmetry of flow ahead of the foil compared to hat aft. The trailing vortices plus the bound vortex nduce a downward inclination of the velocity field; he vertical component is the well-known downwash. ['his causes a tipping aft of the resultant force vector in the foil, which can be attributed to an "induced

drag." As is well-known, this induced drag is, of course, zero for two-dimensional foils and is a mini-mum on a finite wing that has an elliptical spanwise distribution of circulation of the bound vortex.

This concept of replacing the finite wing by bound trailing and starting vortices was first developed mathematically by Prandtl and thus bears his name as Prandtl's lifting-line theory. It should be noted that this theory is applicable only for large aspect ratios and small to moderate loading. For heavily

loaded foils, a _{lifting surface theory is needed.}

Professor Nishiyama uses the lifting line theory in his representation of the wing. He then presents the total velocity potential, including the effects of the free surface. From this he derives the induced velocities which are needed for the determination of r and then the lift and drag.

The results appear extremely useful and the ex-perimental data check nicely with the theory. The real value does, of course, depend on the ease and speed of application to hydrofoil design. This will require a number of calculations to check existing data. In this connection, this paper and the nextone (treating foils having a dihedral angle) should pro-vide excellent working material for the hydrofoil designer.

The project to drill through the crust of the earthProject Mohowill

require a considerable amount of ingenuity to develop the equipment and technique required. The American plan projected presently envisions drill-ing from a marine rig, since the Mohovoricic Discontinuity lies about 16,000

feet below the sea floor in certain Pacific_{areas, while elsewhere, a much} deeper hole would be required. This depth hole, on dry land, would be feasible with present equipment. However, in the Pacific locations, the

ocean is 15,000 to 18,000 feet deep. This means handling unprecedented

lengths of drill stringsover 30,000 feet long. Alternatively, a means for

applying power directly to the drill pipe at the _{ocean bottom may be}

feasible. It appears that a new design drill barge will be required tomeet

the stringent requirements. The Soviet Academy of Sciences is reported

to have set up a new branch in order to solve the necessary problems and be the first to accomplish the project.

from SCIENCE NEWS LETTER June 6, 1959

TETSUO NISHIYAMA

LIFTING-LINE THEORY OF THE

SUBMERGED HYDROFOIL OF FINITE SPAN

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 Mechani-cal 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.

PART II

Hydrodynamical Characteristics of the Submerged Hydrofoil of Dihedral Angle

INTRODUCTION

FROM THE viewpoint of direct application to prac-tical problems, the theory of the submerged hydro-foil of dihedral angle is of extreme importance. Up

to date some theoretical treatments of the

sub-merged hydrofoil of dihedral angle as a three

di-mensional hydrodynamical problem have been

published. These researches were worked out by Tinney and Kaplan; the former' introducesan em-pirical formula for the lift and examines thewave resistance analytically by a rather rough method, and the latter 2 treats of wave patterns and wave resistance. However we cannot find in either paper any consideration of the circulation itself.

Generally the lift is proportional to the circula-tion and the wave resistance to the square of the circulation. Hence in order to examine the lift and wave resistance, it is of primary importance to ob-tain the circulation over the span as precisely as possible, taking the existence of the free water

sur-face into account. It is in this

_{area that} improve-ments still need to be made.

Because of a fairly large aspect ratio and small loading, Prandtl's original concept of the lifting-line may be applied to the submerged hydrofoil of di-hedral angle. Therefore in this paper a calculating method is proposed for obtaining the hydrodynami-cal characteristics of the submerged hydrofoil of dihedral angle from the standpoint of the lifting-line theory.

VELOCITY POTENTIAL

We take the origin of co-ordinate 0 on the still water level with x-axis in the direction of advance, y-axis laterally sideward and z-axis vertically up-ward. Furthermore the ordinate ?I is taken in the direction of span, the dihedral angle of-which is de-noted by 18, as shown in Figure 1.

Applying the concept of the lifting-line, the hy-drofoil can be replaced by a bound vortex, namely lifting-line, and the free vortex may be considered as being situated on the plane parallel to the direc-tion of advance through the bound vortex.

When a hydrofoil of dihedral angle and of half

SUBMERGED HYDROFOILS _NISHIYAMA

span b/2 advances in an infinite fluid with the con- the velocity potential is given by eko . _{After some}

stant velocity C at the distance f from the origin, reduction:

where

w=x cos + (ynm)sin 0

/=sin /3, m=cos /3

In the third and fourth terms we should take the upper and lower terms correspondingly to the posi-tive and negaposi-tive signs respecposi-tively. As the second and fourth terms do not include the variable x, it

may be interpreted that the first and third terms

represent the fluid motion by the bound vortex and the second and fourth terms the fluid motion by the free vortex.

When we denote the velocity potential of the

submerged hydrofoil sb, the boundary condition on

the free water surface can be given by:

0.=Re

sb,=Re

694 A.S.N.E. Journal, November 1959

2 on

ir 00 ch,

sec0 dB e-K('-1-177101"dK

12: r(v)CIN e-n(z+r-17711) COS

u(ynm) du]

0

2

11(

° )r(rt)c1771 tan°

872 So...., -r de.c e-K"-f-17)11)+IK;dK

72h OC

+27r(00h )r(77)(17/10 e-u(z+f-Inl

sin u(ynm)du

K. act. p. ao=0 at z=0 (2)

ax2 az where

Ko=g/C2

Then from (1) and (2) the velocity potential can be expressed in the form:

95=00-, +01+0, (3)

Tood7) sec8 dO e-"" -17111-z) f dK

27i1 e-u(r- Inli-z)cos u(ynm)du

0

(1)

. .

=Re

1C0)mli

co _{e-K(r-17/11-z)} i K

172 iKo b rood??I sec30 dO

0 KKosec20+psec9

where the Rayleigh's viscosity coefficient ix is to be set equal to zero after serving its purpose.

The velocity potential 01 is the image of ch, when the free surface is conceived as a fixed boundary; the first and third terms represent the fluid motion

by the bound vortex at the image point, and the

Now we confine our attention to the induced ve-locity normal to the span. Differentiating normally to the span and evaluating the integral after alter-ing the variable of the integration, we have

2 m2

NO

_{/ (10}

_r(n)

c1 277 w= (Y 77m) 2 ÷ 0

(ynm)

co

[

(77 ) tan° dO K;d1C 87r2 _-a 2

+24

)r(v) ch7e-uf-I

(iiii-o

sinunm)du]

(y

0 00 2 dK I. b_ 2 1 7 3') e-K( f -Nil-0+1K;

-472 K 0 (0 cin secs° sin° dO

-7 K Kosec2O-FiAsecO

lim[452],va,e--f

_r(ochi

_{e-",r-inii) cos u(y-7/m)du}

b M Z/) 0

J;

b 1 2

r

(

JO LT_

T-7 fg ood'? I _{e-u(f-inli-z) sin u(ynm)du}

(6) 0

b

Comparing (6) with (4), we have

lin1[02] wave 2x (the 2nd and 4th terms of ¢,) _(6') From (6') we can see that the velocity potential

of the regular wave is equal to twice that of the

image free vortex at extremely high speeds; in

other words the wave motion just after the hydro-foil approaches gradually to the free vortex motion

INDUCED VELOCITY

second and fourth terms the fluid motion by the

free vortex at the image point. The _velocity poten-tial s62 represents the wave disturbance.

Now transforming 02 by the contour _integration and paying special attention to the vicinity of the hydrofoil, we have this relation at extremely _high speeds:

as the speed becomes higher. The relation _{(6') can}

be used to consider the relation

_{between the} in-duced and wave resistances

_{and the variation of}

the wave resistance with the aspect ratio.

m2 b r (OKI' (Y 27m, f 1,11 Odn '377

-r(n)Kiv(Y-7/m, fInlOch? 47, (4) 71: ° l'(n)kV _{M. f-1911)dn (7)}

NISHIYAMA _{SUBMERGED HYDROFOILS}

where 4(f-17711F-1-(yqm)2 1(2 4(f-177102 nig /

-

ISO( tHnl

[f 1+

4 (f 170 2+ (3' 71m)

T

2 4(f-1,7102+(y-77.1)2}-k---"

2(f inio

1

_8(fini/)2

+2{--

(8)

(yV1)2+4(f-17711)' KOV Ko{(37-77m)2±4(fInI/)) :`/21--1

K.,,v= 2

4(f 170) (ynm)

4K02e-xocr-InIt) r r 1

O(f-10)2+ (y,7m))2 +NRYnrn).2-E4(f In!()2

Ll

4

f-17711 f-17711

Kof (ynm)2+4(f-10)2)

±4V(Y-77m)2-1-4(f-177!W (9)

4(f-10)2

[Ko{(Y'7m)2±4(f-17711)2),/:: it-v= 4(f 17/1/)z+ (Y-77m)2 -Fic02e-Ko(HnIn {4(fInl1)2+(y-77m)12 (Y-72m)2 4{(YV11)2-1-4(f-17711)2)(-1-(2±K2)1

where K, K, and K2 are modified Bessel functions of the 2nd kind, the arguments of which are:

1/21C,{(37nni)2±4(f-17111)21-The terms below the 3rd in (7) are the additive

terms due to the existence of the free surface. Tak-ing the limitTak-ing case, we have:

The circulation on the span of the submerged

hydrofoil is given by:

rn, w(Y)

r(y).t(y)Cla(v)

}, mo=27rw (12)

2 _C

On the other hand the circulation in an infinite

fluid is given by:

ma woo WI

I' co(y)=

_2t(y)C

{

a(y)

(13)

C

f

Taking a fairly large aspect ratio and small loading into account, the 2nd order term may be neglected. Therefore, from (7), (12) and (13) we obtain:

(7,

mo

r(y)=--817.t(Y)[m'i

r(n)Kr(Y-71m, f-17711)thi

A.5.N.E. Journal, November 1959

INTEGRAL EQUATION AND ITS NUMERICAL SOLUTION

lim [the terms below the 3rd in (7)]

= lim [the terms below the 3rd in (7)].. (11)

K0-31.,7)

Therefore the induced velocity normal to the span decreases at extremely low speeds and increases at extremely high speeds. Corresponding to this, the effective angle of attack increases at extremely low speeds and decreases at extremely high speeds.

+1MIb r(OkiiV(37-771n, f-177 I I)C177

--1÷)

±12 1?)

)r(77)Kv(Ynm, f-17711)dn

_h

+r, (y)

(14)

(14) may be regarded as an integral equation with respect to r(y). When the circulation in an infinite fluid is given, the corresponding value in the sub-merged condition can be found by solving this in-tegral equation.

However an analytical solution for this integral

equation is too troublesome to obtain. And so, from

-,1,111.11,11111,-SUBMERGED HYDROFOILS _NISHIYAMA

NISIIIYAMA

the fact that (14) is of Fredholms' type of the

sec-ond kind and the kernal ku(ynm, f 70)

etc.

are regular, we can solve numerically by applying the Nystrom's method.

Hence, putting

I'(y)=2bC y(y), 1'.(y)=2bC y,(y) (15)

(14) can be reduced to

y(y)=.

16rb t(3r;)[2m2ai"Y(77i)Kil (3/J-7711TI, fln, a1" y (7? i)Kliv (y; 771m, f

(37Jnim, fIni11)]

+Y.(37;) (14')

Taking seven points over the span, seven simul-taneous equations are obtained, but we can reduce them to four simultaneous equations owing to the symmetry of the distribution of the circulation. The coordinates and coefficients in the numerical calcu-lation are as follows:

(:)=0.94910

(n2)=-1-0.74153776 ( 713 )=-T0.40584

77, = 0

Thus, as the circulation can be obtained by that procedure, the lift and resistance can be calculated respectively by: aa:: )=0.06474 a., ar,7 a37 =0.19091 a57

)

a47 =0.20898 . (16)

Generally the 2nd order term becomes more or less large at the tip corresponding to the surface-piercing point and its vicinity, and then the accur-acy of the approximation deteriorates. However the

absolute value of circulation is very small at the

tip, and so it may be considered that there is no

decisive effect on the lift and resistance.

NUMERICAL EXAMPLES

Numerical calculations are carried out for the fol-lowing conditions:

aspect ratio A=8

dihedral angle /3= 300 chord lpngth titi== 1.0 immersion fity,o= 2.0

These data 3 were taken from the existing hydrofoil of dihedral angle. The distribution of the effective

circulation over the span is shown in Figure 2;

2bCCaco 1.0 05 ^ -10 SUBMERGED HYDROFOILS 0

Figure 2. Distribution of the effective circulation. (Dotted line shows the corresponding value in an infinite fluid.)

since the dotted line shows the corresponding dis-tribution of circulation in an infinite fluid, the dif-ference between both lines, full and dotted, may be regarded as being caused by the effect of the free surface. Its amount is the largest at the tip which is just at the free surface itself.

The lift is shown in Figure 3; comparing with the corresponding value for the horizontal hydrofoil shown by the dotted line, we can see the effect of a dihedral angle on the lift. It can be safely said that the effect of the free surface becomes more pre-dominant as the dihedral angle is larger and

conse-quently the lift decreases. Further the lift at

ex-tremely high speeds approaches the limiting value which is given by 0.88 Cao, by this present method. On the other hand Tinney's empirical formula

Ca/Cao,

1-0.422e_'

t (19) Empirical formula

----Ca dihedral angle =300 dihedral angle = 0° 1 2 3 4 Figure 3. Lift. 1.0 L=ipC 2b r(77)dn (17) -D= pi r(n)VV (q) dr/ (18) 2 004 002

gives 0.90 C,. From these we can see that by both methods the value is very comparable.

The resistance, which consists of the wave-mak-ing and induced resistance, is shown in Figure 4.

010

0.05

0

1.0

09

698 A.S.N.E. Journal, November 1959

Tinnev

dihedral angle =30°

dihedral angle = 00

3 4

r(Y)=--87t(Y) _ r(n)k-v(y, f17?I)dn+1.7, (y)

where Y 4(f-1771)2 - -EK2e-go"--In0

[

4(f-11)2 {372+4(f-1771)2-2 Kd y2+4 (f 1771)2P Y2

4072-}-4(f)2

As a numerical example, the lift and resistance are shown for aspect ratios 4 and 6 in Figures 5 and 6; the change of the characteristics due to theeffect of free surface is comparatively small. This may be explained by the less prominence of the wave

dis-A=4 6

Figure 5. Lift of the vertical hydrofoil.

At the same time, the corresponding value in an in-finite fluid can be calculated by Kondo's method and is shown by the dotted line parallel to the ab-cissa; the wave-making and induced resistance in-creases by the effect of a dihedral angle and in par-ticular the former is rather predominant. Further the resistance at extremely high speeds approaches the limiting value which is very proximate to Tin-ney's value.

In the meantime, from these considerations it can be concluded that a dihedral angle makes the effect of the free surface predominant and deteriorates the hydrodynamical characteristics of

the

sub-merged hydrofoil.

SPECIAL CASES

Now we consider the special cases from a prac-tical viewpoint.

Vertical hydrofoil

Putting m=0 and 1=1 in the aforementioned

equations, we can obtain the corresponding expres-sions for the vertical hydrofoil. This problem has been examined only slightly by Maruo.5

The integral equation for determining the circu-lation, which is most fundamental, is given by

(21)

turbance. Further, in order to examine this more minutely, the distribution of the effective

circula-0.10 005 Cur J I

A4

A=4

6 3 4

Figure 6. Resistance of the vertical hydrofoil. (20)

1

SUBMERGED HYDROFOILS NISHIYAMA I

Figure 4. Resistance. (Dotted line shows the correspond-ing value in an infinite fluid.)

NISHIYA1VIA

tion is shown for an aspect ratio 4 in Figure 7; the effect of the free surface can be found locally only at the tip, where the circulation is very small, but on the whole its amount is comparatively small. Horizontal hydrofoil

Putting m=1 and 1=0 in the aforementioned

equations, we can obtain the corresponding

ex-pressions for the horizontal hydrofoil which was discussed in detail in reference 6.

WAVE PATTERNS

The kinematical condition at the free surface is given by

84)

-C an

at z -0 ₍₂₂₎

az ax

Now, before entering into the wave patterns caused by the submerged hydrofoil of dihedral angle, we examine the wave patterns of the submerged hydro-foil of horizontal span.

For the sake of simplicity, we assume that the

distribution of the circulation is uniform over the span. Substituting the velocity potential of the sub-merged hydrofoil of horizontal span6 into (22), the

surface elevation can be obtained. But we confine our interest to the regular waves in the rear which contribute to the wave-making resistance.

n/r0/7/C=I see° cosec9 2 7 x 2 2 - ^1r +tan, b/21 2 2

From the 1st and 2nd term in (23) we can see that the Kelvin wave generates at a point just over the tip, and becomes a free wave only somewhat aft.As the speed of the submerged hydrofoil _{is generally} high, the diverging wave is rather predominant. On the other hand, from the fact that the 3rd term in (23) does not include the variable

y, we can see

that the 3rd term has the properties approximate_to the two dimensional wave which propagates in the direction of advance. However from the range of integration the amplitude is larger at the tip than in the middle point of the span. Moreover from the relation

lira (the 3rd term) =lim (the 3rd term) .... (24)

x-00. -co Y-30-2 Go 002 I 1.0 SUBMERGED HYDROFOILS 2bCCaco 004 oh 008 0.1.0

e'"° cos [Icsec2fl x cose+ (y--2 ) sin() d dB rtan-'Y+ 1)/2

j2ir +sec20cosec 9 e-K.,fse"° cos Kosec2q x cos() + (y+-) sin() de 2

sec20 cosec6 e-Kor8"20 cos (K0sec20 x)de

10

-Moire 7. Distribution of the effective circulation over the span of the vertical hydrofoil.

(23)

the amplitude diminishes at considerable distances aft and to the side. Therefore it may be said that

this wave exists only in the vicinity of the

sub-merged hydrofoil and is peculiar to the hydrofoil of finite span.

The surface elevation at extremely high speeds immediately behind ,the hydrofoil is given by:

{ 2+Ylim/ro/c7r=x lira 7ilro/C7r=x

(b +3,\

2 +f.24_ b

_02

(25) ° 2

)

2

)

(25) also expresses the fluid motion by the free

vortex itself. Therefore this shows that the wave

A.S.N.E. Journal, Novaalbar 1959 ₆₉₉

---LI&

.

SUBMERGED HYDROFOILS

motion in the vicinity of the hydrofoil approximates the vortex motion at extremely high speeds.

On the other hand, the corresponding two dimen-sional surface wave 7 is given by:

urnn'iro/Cr -0 (26)

0

x=0

In other words, the two dimensional wave dimin-ishes at extremely high speeds.

(25) and (26) correspond to the properties of the velocity potential and will aid in considering the relation between the wave-making and induced sistance and the variation of the wave-making re-sistance with the aspect ratio.

Now we examine the wave patterns by the sub-merged hydrofoil of dihedral angle. Substituting

(3) into (22), the surface elevation can be

ob-tained; the regular waves consist of the following terms:

undulatory term

sinCOS

Kosec29{xcos() +

(y

+ l ) sine}]

2

[

COS Kosec20{xcost9+

(y-/

)

sine

}]

sin[ b sin[Kosec20 xcos0] cos range of integration Y-F -2--tan-1

tan-1

_x

Thus these show that the Kelvin wave generates not only at the point just over the tip, but also at the vertex of the dihedral angle, and the wave

pe-y-

₂

tan-1

... (27)

(28)

In this paper, Professor Nishiyama has made a significant contribution to the theory of hydrofoils. Reference (1) , Tinney's St. Anthony Falls report of November 1954, is, at present, the most useful re-port on dihedral hydrofoils. It is recommended that the reader scan this report for background

informa-700 A.S.N.E. Journal, November 1959

DISCUSSION

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

culiar to the submerged hydrofoil of finite span has a large amplitude not only at the tip but also at the vertex of the dihedral angle. From this it may be understood why the wave-making resistance be. comes larger in the hydrofoil of dihedral angle than one of horizontal span.

In this paper the hydrodynamical characteristics

for the submerged hydrofoil of dihedral angle,

which is surface-pierced at the tip, are considered in detail.

First, satisfying the boundary condition on the free surface, the velocity potential is introduced and then the fluid motion and the induced velocity are examined. From the assumption of the lifting-line theory, an integral equation is introduced for determining the circulation over the span and then the numerical method is presented for solving this integral equation.

Numerical calculations are carried out and com-pared with the up-to-date results. Lastly the regu-lar waves caused by the submerged hydrofoil of di-hedral angle are considered.

The author wishes to express his gratitude to

Prof. F. Numachi and Prof. S. Fuchizawa for their many cordial guidances.

REFERENCES

E R. Tinney, "Experimental and Analytical Studies of Dihedral Hydrofoil," St. Anthony Falls Hydr. Lab., Univ. of Minnesota, Proj. Rep. No. 41, 1954.

P. Kaplan, "The Drag of Dihedral Hydrofoils below a

Free Surface," Note No. 381, E.T.T. Stevens Inst. of Tech.,

1956.

Billies, "Das Tragfliigelboat," Jahrb. Schiffb. GeselLs. Bd. 46, 1952.

K. Kondo, "Induced Drag of a Monoplane Wing of Di-hedral Angle," Rep. of Faculty of Engineering, Univ. of Kyushu, vol. 15, 1940.

H. Maruo, "Effect of the Water Surface on the Wing." Jour. of Soc. of Nay. Arch. in Japan, vol. 86, 1953. T. Nishiyama, "Lifting-Line Theory of the Submerged Hydrofoil of Finite Span," Part I, JOURNAL or AMERICAN

SOC. OF Nay. Exc., August, 1959.

T. Nishiyama, "Hydrodynamical Investigation of the

Sub-merged Hydrofoil," JOURNAL OF AMER. SOC. OF Nay. ENG., August, 1958.

CONCLUSION

NISHIYAM7A

tion. Tinney, however, used a semi-empirical

ap-proach, which though valuable, does not delve

deeply into the fundamental problem of determina-tion of the circuladetermina-tion distribudetermina-tion. Professor Nishi-yama has very ably filled this void.

NISHIYANIA

hydrofoil by a bound vortex) should be a very good approximation. Most of the envisioned practical hy-drofoil boats using surface-piercing _{dihedral foils}

will have a high aspect ratio

_{and a low loading.}

There is no doubt that

_{a lifting-surface theory}

would yield little of significance over the lifting-line theory.

Professor Nishiyama utilizes the _{same methods} and procedures that he developed in earlier papers on hydrofoils for this JOURNAL. This is summarized succintly in his conclusions.

It is interesting to note that the dihedral hydro-foil is a very general hydro-foil. When the dihedral angle, /3, is zero, we have the fiat submerged foil; with we have the vertical strut at angles of at-tack. Both cases are discussed in this paper. Con-siderable work has been done, of course, by Wu, Nishiyama and others on the fiat foil, but very little exists for the vertical foil. This latter case is impor-portant. One or two vertical struts are used to sup-port the flat foils on some hydrofoil _{boats. It is}

im-portant to determine the side forces

_{exerted by} these struts in a turn. In many cases, the rudder is

a moveable portion of one of the

_{-vertical struts.} The lift and wave-making resistance are of

inter-A theory that the universe is hot, instead of cold, has been proposed by scientists from Harvard University and Cambridge University. The hot

uni-verse theory contains an explanation for puzzles_{such as the origin of cosmic}

rays, galaxies, and a galactic radio glow which _{has been observed. This} theory holds that space is filled with hydrogen atoms having a velocity

equivalent to a temperature of about one hundred million degrees, but ina

concentration of one or two per cubic yard of space. Compared with these

energetic atoms, the star galaxies themselves are cold. As such _galaxies

speed through _{space, leaving a cold pocket in their wake, the hot}

inter-galactic hydrogen condenses. _{This process is supposed}

as possible for the birth of new galaxies. Professors Thomas Gold and Fred Hoyle, originators

of the theory, believe that hot _{hydrogen clouds surrounding}

galaxies have

sufficient energy to emit radio _{waves which have been observed.}

from SCIENCE NEWSLETTER July II, 1959

A.S.N.E. Journal, Novimber 1959 ₇₀₁

SUBMERGED HYDROFOILS est, but perhaps more important are spray resist-ance, cavitation and ventilation. These _problems will all be considered in experimental _programs

presently underway at the

_{David Taylor Model}

Basin and elsewhere.

It is interesting to _{note the comparison between} Tinney's and Nishiyama's results, shown on Figures 3 and 4. Tinney treated just the case of high Froude number (high C and small f). There is excellent agreement at Froude numbers above _{4. Nishiyama}

has shown though, that

_{this formulation}

_{can be}

considerably in error at Froude numbers of about 2, which may be of importance in the take-off con-dition.

On the basis of the results of this paper, it ap-pears that the smallest dihedral angle is the most desirable. However, _{as Tinney points out, a large} dihedral angle is _{very useful in order to avoid} sep-arated flow at low angles of _attack.

This paper by Professor Nishiyama _should pro-vide excellent guidance for the extensive experi-mental program that is underway on the dihedral hydrofoil. It is an excellent example of the concen-trated effort being made in Japan on the theoretical problems of wave resistance.