Aspects of hydrofoil design: with emphasis on hydrofoil interaction in clam water

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HANS JORGEN BJELKE MORCH

ASPECTS OF HYDROFOIL DESIGN;

WITH EMPHASIS ON HYDROFOIL

INTERACTION IN CALM WATER

TECHNISCHE _UNIVERSITET Laboratorium voor epshydromecharka chief ioe eg _{28 CD Delft} .' 15- 7888 _{015 -78183e}

N

DOKTOR INGENIORAVHANDLING 1992:9

INSTITUTT FOR MARIN HYDRODYNAMIKK TRONDHEIM

UNIVERSITETET I TRONDHEIM

NORGES TEKNISKE HOGSKOLE _{MTA-rapport 1992:86}

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Aspects of Hydrofoil Design;

with Emphasis on Hydrofoil Interaction

in Calm Water.

by

Hans Jorgen Bjelke Mach

Division of Marine Hydrodynamics

The Norwegian Institute of Technology

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To Line for patience and encouragement

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ABSTRACT

In the design of the foil system on a hydrofoil craft it is important to minimize resistance and avoid cavitation. This requires thorough knowledge of the flow conditions on the hydrofoils. The inflow at the rear foil can be greatly affected by the presence of one or more front foils,

with great spanwise variations.

A method to calculate the direct and the indirect problem of a hydrofoil is discussed. Changes

in inflow due to the presence of other hydrofoils are accounted for.

To calculate the induced flow at a rear foil, two methods have been investigated. One is based on

linear three-dimensional theory valid when the free vortex sheet is not strongly rolled up, like

behind large aspect ratio hydrofoils. It is shown both by theory and through model experiments, that one has the possibility to place the rear foil in an upwash resulting in reduced total drag.

Behind small aspect ratio front foils the free vortex sheets will have rolled up into pairs of counter-rotating vortices at the position of the rear foil. Therefore, in the second method the

rolling up of the free vortex sheets of the front foils are calculated before the change in inflow at

the rear foil is calculated. For the foil configuration discussed here, a comparison with a

simplified method in which the free vortex sheets of the front foils are assumed fully rolled up show good agreement.

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ACKNOWLEDGMENTS

This work has been financed by activities in a research project on high-speed vehicles under the

Royal Norwegian Council for Scientific and 'Industrial Research and a scholarship from

Norwegian Fisheries Research Council.

II would like to thank my supervisors Knut JL Minsaasi and Odd 'Mt Fatinsen for valuable.

guidance and encouragement

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NOMENCLATURE

A b2 , aspect ratio tproj

Aproj = horizontal projected area of the foil

foil span

distance between the two line vortices of a fully rolled up free vortex sheet

= foil chord

drag coefficient profile drag coefficient

CDProfIle

F0.075 ITTC 1957

(logRn-2) 2

section force coefficient lift coefficient

section lift coefficient drag

Di = induced drag

Dpro file = profile drag

time step used in Euler integration

Ei(r)

= exponential integral

total force of foil side force of foil front foil

rear foil

Froude number, based on depth

gh = Green function = acceleration of gravity

H(s)

= Heaviside function = foil immersion imaginary unit

/w

= contribution from free vortex sheet to velocity potential, see chapter 3.3

= unit vectors in Cartesian coordinate system

ko Tyr , wave number foil lift

'positive integer

= unit normal vector to a surface, positive direction into the fluid

rife

= number of free surface elements, two-dimensional theory

nseg = number of elements or horsehoe vortices used to represent the hydrofoil

nvor = 2N + 1, number of point vortices representing the free vortex sheet nwe = number of dipole elements representing the free vortex sheet

= pressure Rn

=t4,

, Reynolds number = (Y 71)2 + R0 =

71)2 +

+ (Y 77)2 C f n = CL = ci = j , k = CD = = FN =

FF

RF

Fnh

Ox 02

(z

02 02 +

(z 02 =

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control surface

control surface at infinity

small cylindrical control surface around the field point control surface far down in the fluid

control surface consisting of the free surface

control surface including both sides of a free vortex sheet = semispan of foil or of foil segment

Sh = horizontal component of foil semispan

= spanwise coordinate , = ,

()

= time

= foil thickness

= speed of advance [m/s], corresponding to V, (u, v, w) = induced velocity components

(Au, Ay, Aw) = induced velocity from front foil(s)

= induced velocity vector, two-dimensional theory

V, = ship speed [knots], corresponding to U

(x,y, z) = field point

a

= angle of attack

dihedral angle, angle of inclination to the horizontal plane, see fig. 2.2.2 zero lift angle

zero lift angle in an infinite fluid = artificial smoothing parameter

circulation

F0 = mid-span circulation

kinematic viscosity

= Green function used in Treffzt plane analysis, see chapter 3.

= fluid density c) = singularity point

= wave elevation

= 01 + 0, total velocity potential

01 = incident undisturbed potential

(/) = perturbation potential

0+ = jump in the velocity potential across the free vortex sheet

am k

a

7 ,1t) = induced velocity components

az ay 0: 7 = Soo = = SB = SF = Sv =

0

= 0o = 0o_ = =

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1. Introduction

This thesis deals with theoretical and practical aspects of the design of hydrofoils and the interaction between the foils on hydrofoil ships. The typeofhydrofoil ship considered here is

the so called hydrofoil catamaran.

The aim of the present report is to discuss and to provide means for calculation of lift and drag characteristics of the hydrofoils on a hydrofoil ship.

On hydrofoil catamarans speeds up to 50 knots are actual. At such speeds cavitation is a critical factor which may reduce the lift, cause problems of erosion and increase the resistance.

In order to avoid the problems of cavitation a detailed knowledge of the flow conditions at the foils is necessary. The interaction between the foils must be accounted for. Typically, the presence of one or more front foils affect the otherwise undisturbed inflow at the rear foil.

Methods to calculate the resulting inflow at the rear foil are discussed.

The applied hydrofoil theory and numerical methods are based on the assumption that the

catamaran hulls are out ofwater and thus have no effect on the flow. Only operation in calm sea with uniform forward speed is considered.

On hydrofoil catamarans a variety of different types of foilsand configurations have been

proposed. In our analysis two foil configurations are dealt with, box-foils in tandem and two T-foils up front and a box-foil as the rear foil, see figures 1.1 and 1.2.

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The T-foil is characterized by a plane lifting surface with a symmetric vertical strut at the

mid-span connecting it to the hull. The 1-foils considered in this report have aspect ratios about 3.

The box-foil is a plane foil with a span equal to the beam of the catamaran with vertical struts at or close to the wing tips connecting it to the hulls.

Front

vertical strut

9

Rear Foil

Fig. 1.2. Foil configuration for 120 to 200 ton hydrofoil catamarans.

This thesis has two main parts, chapter 2 and chapter 3.

In chapter 2 the hydrofoil is analyzed following the Prandtl lifting line theory in which foils of Foils'

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large aspect ratio is replaced by a lifting line. The lifting line contains all the bound circulation of the hydrofoil integrated along the chord and has the same spanwise lift distribution as that of the real hydrofoil.

The velocity potential of a lifting line with arbitrary dihedral angle or angle of inclination to the

horizontal plane, in uniform flow under a free surface is discussed. The influence of the free surface on the flow both on the hydrofoil itself and downstream, is discussed. A numerical method to calculate the direct and indirect problems of a hydrofoil is discussed. In the direct

problem the foil geometry is given and the spanwise load distribution is calculated. The indirect problem is the opposite, a load distribution is specified and the foil geometry is calculated. This is discussed in more detail in chapter 2.5. The numerical method takes into account the change

in inflow on a rear foil due to one or more front foils. Results from model experiments with

hydrofoils in tandem are presented and compared with numerical results.

Different numerical methods are discussed in chapter 3 to account for the rolling up of the free vortex sheets of the front foils. This is a most relevant situation for the second foil configuration

with the two T-foils up front. The numerical method is based on Treffzt-plane analysis and

allows for the effect of the free surface. Also allowed for are the effects of interaction between

the neighboring free vortex sheets originating from the two front foils and from an assumed load distribution on the rear foil, on the evolution of the free vortex sheets and the resulting

inflow at the rear foil.

A paper on some of the problems treated in this thesis has recently been presented, MORCH

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2. Hydrofoil Theory

2.1 Introduction

In the following potential theory has been applied to calculate the lift and drag characteristics of a hydrofoil and the flow field behind it, to determine the working conditions for a rear foil. The

basis for this theory is the velocity potential of a lifting line with general dihedral angle, 3, in uniform motion under a free surface. The author is greatly indebted to professor Faltinsen for

valuable guidance throughout this work.

A similar velocity potential has been presented earlier by WI.J (1954) for plane lifting lines.

NISHIYAMA (1965) presented a lifting line theory for hydrofoils with general dihedral angle.

The velocity potential which is applied here is based on a Green function derived by

NEWMAN (1987), making it better suited for numerical analysis than earlier expressions.

KAPLAN ET. AL. (1960) used Wu's expression for the velocity potential to calculate the

vertical induced flow behind plane hydrofoils. In our thesis the presented velocity potential can

be used to calculate both sidewash and downwash at arbitrary points in the fluid. A numerical method based on a discretized lifting line theory for aerofoils presented by BLACKWELL

(1969) has been adopted to calculate the direct and indirect problems of a hydrofoil. Changes in inflow conditions due to the presence of one or more front foils are accounted for.

The applied theory and numerical method are compared with model test results from the towing

tank at MARINTEK for two foil configurations. Two box-foils were run in tandem at varying

speed, immersion and distance between the foils. In the other configuration two T-foils were up front and a box-foil was the rear foil.

2.2 The Velocity Potential of a Lifting Line with General Dihedral

Angle in Uniform Motion under a Free Surface

2.2.1 The Velocity Potential

We consider a hydrofoil which moves forward with constant speed and fixed immersion. The coordinate system is defined in figure 2.2.1. It is fixed with respect to the hydrofoil. Origo is in the plane of the undisturbed free surface with the z axis positive upwards. The I axis is positive in the downstream direction. U is the undisturbed incoming flow equal to forward speed and it is directed in the positive a-direction. The flow is thus independent of time.

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Figure 2.2.1. The coordinate system

The hydrofoil is represented by a lifting line containing the total bound vorticity. We define x = 0 at the longitudinal position of the lifting line. We impose the restriction that the lifting line has no sweepback, ie. x = 0 along the span of the lifting line. The free vortex sheet downstream of the lifting line contains the shed free vorticity. The far-field view of the hydrofoil is shown in figures 2.2.2 and 2.2.3.

-S

Figure 2.2.2. Far-field view of hydrofoil projected on the y-z plane.

-op-

foil far-field Projection of the x-z plane behind the hydrofoil.

Figure 2.2.3. Projection of the lifting line and the free vortex sheet on the x-z plane. Z

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The total velocity potential in the fluid domain outside the free vortex sheet can be written:

(I) = çb1 + (2.2.1)

where 01 is represents the undisturbed incoming flow and 0 is the unknown velocity potential. In our case of a hydrofoil in uniform motion we can write:

=Ux

(2.2.2)

The velocity potential cl) of the flow must satisfy the following conditions:

Outside the free vortex sheet region non-rotational fluid is assumed and the Laplace equation must be satisfied:

72,b (2.2.3)

On the mean free surface the dynamic and kinematic free surface conditons must be satisfied. The kinematic free surface condition states that the particles at the free surface remain at the free surface:

Ua((r,Y)

= 0 on z = 0

az

ax

The dynamic free surface condition expresses that the pressure at the free surface is constant equal to the atmospheric pressure:

- g ( x , y) = 0 on z = 0

ax 520

_{, ao}

u on = 0 ax2 az k, _u2 wave number 13 (2.2.4) (2.2.5)

Both kinematic and dynamic free surface conditions have been linearized about the mean free surface, z = 0, since the induced fluid velocities are assumed to be small quantities compared to the forward speed U. This may be a poor assumption when the foil submergence is low, since the induced velocities due to the foil show a singular behaviour at the wing tips. This will also be a problem for surface piercing hydrofoils.

The linearized kinematic and dynamic free surface conditions can be combined to give:

(2.2.6)

The radiation condition must be satisfied. From this it follows that there are no waves far

upstream of the hydrofoil.

On the hydrofoil body, the boundary condition, ie. no flow through the foil, must be satisfied:

'act,

0 (2.2.7)

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The velocity potential is discontinous across the lifting line and the free vortex sheet, and there is a jump in our unknown velocity potential equal to 0+

Applying Green's second identity the unknown velocity potential at a point (x, y, z) in the fluid, can be written as:

ac

a

47-0(x, y, z) =

f f (0+

)On"'dS + fs

an

Gcb

an

)dS

sv

,

SF = mean free water surface

Sv --= free vortex sheet behind the lifting line G = Green function

n+ = normal vector, see figure 21.1 for meaning of + and -n = -normal vector, positive i-nto the fluid

For the Green function which is the potential of a source in steady motion under a free surface, we use an expression derived by NEWMAN (1987). Newman's expression is better suited for numerical integration than the conventional form presented by WEHAUSEN and LAITONE

(1960), equation.(13.36).

1 1

L0[2

G(x, y,z;C,q,() = Re{

R0 cos Bey Ei(v)d8

2

+i4k0H(ko(x e))

f see 6eud9)

(2.2.9)

Where

v = ko(z + () cos' + ko ly

771 cos sin + iko I cos

u = ko(z + C)sec2 0 + iko IY /71 sec' 0 sin 0 + ikolx sec 0

R=

(Y +(z+ C)2]

Ro = 6)2 + (1/

+(z - C)2l4

Ro is the distance between a singular point on the lifting line or on the free vortex sheet and a point in the fluid, R is the distance from an image of the singular point mirrored about the mean free surface, z = 0, to the point in the fluid. El is the complex exponential integraldefined in

ABRAMOWITZ M. and STEGUN LA. (1964) and H is the Heaviside step function. Itshould

be noted that the expression given by NEWMAN J.N. (1987) has been altered to fit into our system of coordinates.

It is important that the Green function has the following properties; it must satisfy the linearized free surface condition, eq. (2.2.6). Also, it must satisfy the radiation condition and it must die out when the distance between the singularity point and the field becomes large:

(2.2.8) 1 G [(x )2 + (y 77)2 + (z + ()21 , as)? oo (2.2.10) = e

- e)2

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The second term in equarion.(2.2.8) can be written in the following manner

irsp (O-STG,

GP) dS

_n

=

_ffSt

(Or'

Cra4) dS

crC

=

fro. fre. (09 GV)

ckdq = f 70. (f rop (cl) de)

_.L1r00

_([41

_{chi = 0}

(2.2.1_1)

Thus we can write::

stwO(x, y, z)

r

(_ sin + cos

)dede

(2.2.12)

JO an

where I' = 0+

Referring to equation (2212) we need to evaluate .17, ;,9,-? de and' Lc° Vde

We define: F.= Gde 0 which we write: 1 = 1.2 + 12 + 13 K2.2.136)

Ii = i

1

)de

1 (2.2.130 o Ro R .1

12 = Re{i2k0

I 2

cos 9

I

evED(v)ded0} (2.2.13d)

if-1 0

(00

.13,= Re ti4ko

sec2 j

H(ko(x e))e"dedr9)

2

0 Our expression for the velocity potential; is now:;

1 is

al,

34 #(.,yi,zy.

Re{

)s

0-1-(--1---1--)cosfilds}

(2.2.14)

47r ay,

az? 317a( a(

a(

Thus we ;needti

"t n = 1,2,3 and an- n = 1,2,3. In appendix A the complete set of expres-

_ao

sions are derived by analytical integration over the free vortex sheet for e = [0, co]. When doing the integrations it is assumed that the free vortex sheet remains constant in the same plane as that of the lifting line.

2.2.2 Physical] Interpretation of the Velocity Potential

In equation (2.2.12) the velocity potential of the hydrofoil is made up of the velocity potential of a lifting line in infinite fluid, cfr. appendix A and 'TRUCICENBRODT (1953) , its biplane (2.2.13a)

(2.2.33e) =

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image about the mean free surface and a gravity dependent part which we will call the wave part. Hence:

= Olift.line inf.fluid Obiplane image + 0wave part (2.2.15)

Referring to Equations (2.2.13) and (2.2.14) the lifting line in infinite fluid is represented by the ri and ( derivatives of the first term in 11 and the biplane image is represented by the derivatives

of the second term. The wave part of the velocity potential is the sum of the ri and derivatives of 12 and 13:

f time inf . fluid =

a (So' Thd)

a (focc ;hide)

sin,3

a(

cos/3j de (2.2.16a)

(2.2.16b)

(2.2.16c)

Obiplane image

=If r

(fo'

i+d),iio

a (fcco kd) cos/31

de

47r

wave pare = ite

- - sin

(- - cos

Pi a,s-)

47r

a,

ac,

a(

1 813, . ,812

ar,

, ,

The velocity potentials of the lifting lines are discontinuous across their free vortex sheets. Since we restrict ourselves to be in the water, z < 0, we only have to bear in mind thediscontinuity

across the free vortex sheet of the lifting line in infinite fluid

2.2.3 Asymptotic Behaviour of the Velocity Potential

In the following the behaviour of velocity potential at low and high forward speeds both at the hydrofoil itself and downstream is discussed. A behaviour similar to the one discussed here has been observed by eg. KAPLAN ET AL.(1960).

In the vicinity of the foil the linearized free surface condition, equation (2.2.6), dictates the

behaviour.

At low speed, /co cc; equation (2.2.6) becomes:

= 0 at z = 0 (2.2.17)

that is, the mean free surface act as a wall. Thus, the velocity potential becomes that of a foil in

infinite fluid plus its wall image about z = 0. The circulation of the wall image is in opposite direction of the circulation of the foil in infinite fluid.

For high speed,k0 4 0; equation (2.2.6) becomes:

= 0 at z = 0 (2.2.18)

The velocity potential is now equal to that of a foil in infinite fluid plus its biplane image.

+

a<

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This is illustrated in figure 2.2.4. The velocity potential as defined by equation (2.2A2), has been calculated as a function of speed of advance at the mid-span position of an elliptically loaded, plane hydrofoil with high aspect ratio and operating close to the free surface. Due to

the discontinuity of the term due to a lifting line in infinite fluid of the velocity potential across the hydrofoil and the free vortex sheet behind it, only the biplane image and the wave part are

shown.

At low speeds of advance the value of the wave part is twice the value of the biplane contribution, but with opposite sign. Hence, the mean free surface acts as a wall.

At high speeds however, the wave part becomes increasingly negligible and the hydrofoil acts as the lower wing in a biplane configuration.

0.4

-0.3 0.2 Ef 0.1 -o wave part

_________

_ biplane image 17 I- I I I 2 4 6 8 10 Fn, Fig. 2.2.4.

Influenceofforward speed on the parts of the velocity potential which represent

the effect of the free surface, calculated at the mid-span position of a plane

hydrofoil with elliptical load distribution.

Far behind the hydrofoil, x + op, the physical requirement is that the surface should be smooth. Hence, we should expect the mean free surface to act as a wall. In figure 2.2.5 the biplane and the wave part of the velocity potential are plotted as functions of downstream distance for both low and high forward speed.

As seen from figure 2.2.5, the mentioned physical requirement is fulfilled at both very high and

0.0 -0.1

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very low speeds. The difference between them is that at high speeds there are local waves which die out far downstream of the hydrofoil. At intermediate speeds of advance it is possible to see the regular wave system which comes into play once the dominant local waves have died out. It can be seen in figure 2.2.5 that the mean value of the wave part of the velocity potential once the local waves have died out, oscillates about the value of minus twice the value of the biplane part of the potential.

1.2 1.0 0.8 0.6 0.4 _Fnh=10.43 Frih=3.91 Fnh=0.26 wave part biplane image 0.2 0.0 -0.2 0 10 20 _xis 30 40 Fig. 2.2.5.

Influence of downstream distance on the parts of the velocity potential which represent the effect of the free surface, calculated at the centerline of the free vortex sheet

behind a plane hydrofoil with elliptical load distribution. h/s=0.18.

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2.3 The Induced Velocities from a Hydrofoil in Uniform Motion

In this chapter we will attempt to clarify the relative importance of the different parts of the

velocity potential on the induced velocities both on the hydrofoil itself and in the fluid

downstream of the hydrofoil.

The sideways and vertically induced velocities induced by a lifting line in steady motion under a

free surface are found from the y- and z-derivatives of the velocity potential in equation

(2.2.14):

a0(x,y,z)

r

ay

4r

fa2ii a2,2 a2,3 a2ii a2,2 _a2,3

ds

larray+anay+anayrin Plgay+

gay

'gDy)c°s

The terms in equations (2.3.1a) and (2.3.1b) are listed in Appendix B.

ao

In the following v is used for Ty and w for

In the same manner as in chapter 2.2.2, equations (2.2.15) and (2.2.16), the induced velocity

due to the hydrofoil is regarded as the sum of three parts; the flow due to a lifting line in infinite fluid, its biplane image and a wave part. The first two are contained in the !j-terms and the wave part in the 12- and /3-terms.

The term due to the lifting line in infinite fluid causes a problem when the field point is exactly

at the lifting line itself or in the plane of the free vortex sheet or the wake, behind it. When

integrating along the span this expression has a singularity. An attempt to avoid the problem of

the singularity has been done by dividing the numerical integration into two regions on either side of the singularity and to use a numerical routine that is able to handle singularities at the

end points. However, numerical overflow still occurs.

Therefore, the solution throughout this chapter has been to move the field point a small distance

off the plane of the lifting line and its wake. This we can do since the velocity normal to the plane of the lifting line is continuous across the plane. Figure 2.3.1 shows the influence of

increasing the distance between an elliptically loaded foil in infinite fluid and the field points.

From classical theory eg. PRANDTL and TIEDENS (1934), one finds that the elliptically

loaded foil has a constant downwash along the span given by:

19

(2.3.1a)

(2.3.1b) a(txx,y,z) I _a2/, _+a2/2 _{a2r3 lc,} _a2r, _a2/2 _a2/3

ds*

az 47r

fs

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-1.5

v4s)=

OF

4s

In the case of the foil in figure 2.3.1, equation (2.3.2) gives a constant downwash of 0.25Prils]

for a unitro.The effect of calculating the downwash above the foil is most pronounced towards the wing tips.

1-0.245, vdist/s=.021 vdistis=0.005 vdist/s=0.01 - - vdist/s=0.02 Eq. (2.3.2) 1-0.2473, vdistis=.011 1-0.2481, vdist/s=.0051 Fig. 2.3.1.

Self-induced downwash on an elliptically loaded foil in infinite fluid along the span

at different small vertical distancesover theliftingline.

Figure 2.3.2 shows the different components of the downwash at various depths above and

below the same foil as in figure 2.3.1 when operating at 50 knots with an immersion of 2.0(m].

(2.3.2) LI.' 17; 1.0 0.8 0 6 0.4 0.2 0.0 -0.2-/ -1.0 -0.5 0.0 0.5 1.0 1.5 s-/s

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-...

-0.05 -0.10 7, -0.15--0.20 -0.25

...

lift.line int.fluid --- biplane image - - wave part total vel.p0t.

...

-0.22 -0.26 -0.28 -2.04 -2.00 -1.96 Fig. 2.3.2.

Self-induced downwash from the different components of the velocity potential of an elliptically loaded hydrofoil at the mid-span position at various depths.

his=2.0.

In this chapter a "safe" vertical distance of 0.01[m/ above the lifting line has been used which

gives a satisfactory result at least at the mid-span position.

To illustrate the relative importance of the different parts of the velocity potential we have in the following shown plots of the induced vertical velocity at the mid-span position of a plane lifting line as function of forward speed.

Calculations have been performed for lifting lines with semispans ranging from 1.25[m] to 10.0[m] at an immersion equal to 2.0[m], to study the effect of paramaters like immersion and lifting line span ratio, see figures 2.3.3 and 2.3.4. Assuming that the chord lengths of the

hydrofoils are from 1.01ml to 2.0[m] this gives aspect ratios ranging from 2.5 to 10.

To verify the calculations the vertically induced velocities are calculated directly using equation

(2.3.1b) and through linear approximation using the expression for the velocity potential, equation (2.2.14). The calculated velocities are the same except at low Fnh-numbers for the hydrofoil with the greatest semispan as seen in figure 2.2.4. Numerical problems occur in the wave part of equation (2.3.1b). The expression for the velocity potential is less subject to

numerical problems and it is seen the linear approximation gives the correct asymptotic value as Fnh 40, which is minus two times the biplane images such that the free surface acts as a wall.

-030

1

I I I -4 -3 -2 -1 0 Vs -0.24 Fnh=5.81,,

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0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 2 4 6 8 10 Fig. 2.3.3.

Vertical self-induced velocity at the mid-span of a plane elliptically loaded hydrofoil as a function of Speed, calculated directly using eg.(2.3.1 by and by linear approximation using the velocity potential, eq.(2.2.14). tvs=1.6.

Linear approximation, eq.(2.2.14)

Direct calc. eq.(2.3.1b)

4 Vane inf.fluid --biplane image - -wave part total vel.pot. 8 Fnh6 10 Fig. 2.3.4.

Vertical self-induced velocity at the mid-span of a plane elliptically loaded hydrofoil as a function of speed, calculated directly using eg.(2.3.1b) and by linear approximation using the velocity potential, eq.(2.2.14). h/s=0.2.

As can be expected the effect of the free surface, ie. the sum of the contributions due to the wave

part and biplane image, becomes increasingly important with decreasing his values. However,

0.00 -0.05

f.f -0.10- _{----biplane Image}litt.lineint.fiuid

- -wave part total vel.pot.

-0.20 -0.25

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the contribution from the wave part approaches zero with increasing forward speed, but more slowly for small his values. The governing geometric factor is the value of his. From figure

2.3.4 one may conclude that for hydrofoils of moderate to high aspect ratios, ie. higher than e.g.

5, operating at speeds ranging from 40 to 50 knots, the wave part must be taken into

consideration when designing the hydrofoil. figure 2.3.3 shows that one can neglect the wave part for small aspect ratio hydrofoils operating at the same speed, which does save a lot of

computing time.

In figures 2.3.7 and 2.3.8 the influence of forward speed and his on the downwash along the

span is shown for elliptically loaded hydrofoils.

However, behind the hydrofoil the wave part of the velocity potential of the lifting line under the free surface will approach the value minus two times the value of the biplane image as discussed in chapter 2.2. As seen in figures 2.2.5 and 2.2.6 for a given hydrofoil the distance downstream

when this value is reached decreases with decreasing Froude number. This of course, also

applies to the induced velocities.

0.0 -0.1 -0.4 -0.5 inf.fluid ----biplane image - - wave part total vel.pot. I I- I I I I I I 20 40 60 80 100 120 140 160 xis Fig 2.3.5.

Vertically induced velocities in the centerline of the free vortex sheet behind a plane

elliptically loaded hydrofoil as a function of downstream distance, calculated directly using eg.(2.3.1b) and by linear approximation using the velocity potential,

eq.(2.2.14). Fnh=5.81, h/s=1.6.

23

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1.0 0.5 0.0

...

-0.5 -1.0 -1.5 xis Fig 2.3.6.

Vertically induced velocities in the centerline of the free vortex sheet behind a plane

elliptically loaded hydrofoil as a function of downstream distance, calculated directly using eg.(2.3.1b) and by linear approximation using the velocity potential,

eq.(2.2.14). Fnh=5.81, h/s=0.2.

Figures 2.3.5.and 2.3.6 show the influence of foil geometry, his, on the vertical induced

velocities behind the hydrofoil for an operating speed of 50 knots and a given immersion of 2

meters.

For his=1.6 the effect of the free surface is small and the downwash from the wave part approaches the value minus twice the downwash from the biplane image both sooner

downstream and after less fluctuation due to the local wave than is the case for h1s=0.2. For hls=01 the effect of the free surface is strong. With decreasing his the downwash from the

biplane image will be as strong as the downwash from the lifting line in infinite fluid. Also the wave part becomes more dominant with stronger fluctuations before the effect of the local wave

dies out. If the hydrofoil in figure 2.3.6 were a front foil, it can be observed that a rear foil

would operate in a significant upwash at distance of seven semispans downstream.

The Importance of the Different Parts of the Velocity Potential on the Flow Field at the Lifting Line Itself

Figures 2.3.7 a,b and c and 2.3.8 a,b and c show the relative importance of the different parts of the velocity potential on the spanwise downwash on plane elliptically loaded lifting lines under a

free surface for various semispans ranging from 1 to 10 [m], at a constant immersion of 2[m], ie. his ranging fom 0.2 to 2.0. Forward speeds are 30 and 50 knots or Fnh=3.48 and 5.81.

Mine inf.fluid

----biplane image - -wave part

total vel.pot.

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At Fnh=3.48 it is seen from figure 2.3.7a that a hydrofoil with hls=2.0 operates almost as if it were in infinite fluid. With increasing semispan both the biplane image and the wave part

become more important However, the importance of the wave part increases also relative to the biplane image with increasing semispan.

For a hydrofoil with a large semispan as in figure 2.3.7c, the effect of the surface is highly noticeable in that the sum of the downwash from the biplane image and the wave part at the mid-span position is higher than the contribution from a lifting line in infinite fluid. Towards

the wing tips it is less due to the behaviour of the biplane image.

The effect of increasing forward speed to Fnh=5.81 is a reduction of the importance of the wave part relative to the lifting line in infinite fluid and its biplane image. As discussed previously the wave part approaches zero when the speed goes to infinity, Figures 2.3.8 a,b and c show that the effect of the wave part on the the lifting line itself may be ignored for hydrofoils with his greater

than 0.4, as in figures 2.3.8 a and b. For hydrofoils with larger semispans the wave part may still be important, as in figure 2.3.8c in which the downwash due to the wave part amounts to

9% of the total downwash at the mid-span position and even more towards the wing tips.

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Figures 23.7 a,b and c.

Influence of the different parts of the velocity potential on the self-induced downwashi along the span of plane elliptically loaded hydrofoils with different semispans, operating at 30 knots and at an immersion of 2 [in].

The influence of the different parts are indicated at the mid-span in per cent.

2.3i.7a. Fnh=3.48, h/s=2.0. 0.6 0.4 o 02 _9.5% LT) 0.0 4e-- I .. .. .. ..

...

eKevh, Fig 2.3.7b. Fn =3.48, h/s=0.4: -0.5 atline nf fluid ---- biplane image - -wave pan total vel.pot. 0.0 0.5 11.0 1,5 -0.5 0.0 10.5 10 iskVer Fig. 2.3.7c. Frh=3.48, h/s=0.22 6.6 0.4 02 cf. 13' 0.0 -02 -0.4 -0.6 0.6 0.4 0.2 0.2 0.4 0.6

0.4 0.6 -100% 0.6% 95.6% 100% -1.0 -0.5 0.0 0.5 1.0 /s

--

_22.7%

--

_100% I I I -1.0 I I I -1.0 s./s Fig.

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Figures 2.3.8 a,b and c.

Influence of the different parts of the velocity potential on the self-induced downwash along the span of plane elliptically loaded hydrofoils with different semispans, operating at 50 knots and at an immersion of 2 [m].

The influence of the different parts are indicated at the mid-span in per cent.

0 6 0.4 -o 0.2 73 0.0 -0.2--0.4 -0.4 -0.6 0.6 0.4 -, 0.2 -t-z Q.) 0.0 0.20.4 0.6 0.6 0.4 -o 0.2 -ET, 0.0 0.2 -0.4 -0.6 --0.2% 3.0% 96.8% 100% ----biplane image -wave part -total vel.pot. 3.5% ' 26.5% 70% 100% I I I I I -1.0 -0.5 0.0 0.5 1.0 s-/s Fig. 2.3.8b. Fnh=5.81, his=0.4. 9.0%

,

100% I i I I I -1.0 -0.5 0.0 0.5 1.0 s-/s Fig. 2.3.8c. Fnh=5.81, h/s=0.2. 27 -- ---I f I -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 s-/s Fig. 2.3.8a. Fnh=5.81. h/s=2.0. lift.line inf.fluid

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35.2%-2.4 Induced drag

The induced drag and the lift on a hydrofoil section follow from pressure integration over the

foil section:

.Rs*)=

-if

pita

15 FS(s*)

where SFS(s*) is the surface of the foil section and the unit normal vector is positive into the

fluid.

Using the Bernoulli equation equation (2.4.1) is expressed by: -

1,11

(v-2 _{j isFs(s*)}

The spanwise increments of lift and induced drag are given by:

dL(s)=s)-

(2.4.3)

(ID As* ) Rs* ) (2.4.4)

The inflow= a foil section is given by:

= [J, v+dv, w+Aw] (2.4.5)

where v and w are self-induced velocities at a position on the lifting line which represents the

hydrofoil, and given in equations (2.3.1a) and (2.3.1b) respectively for x=0. dv and dw

represent the change in inflow due to one or more front foils.

Keeping only the linear terms and assuming that the bound vorticity along the chord is

concentrated in one point equation (2.4.2) gives:

dL(s*)= pU Tcos 13 (2.4.6)

do

* ) = pr[(v+dv) sin - (w+dw) cos131 (2.4.7)

where the term in brackets in equation (2.4.7) is the normal velocity to the span at the foil

section.

The total lift and induced drag follow from integrating along the span, including all end plates or struts: s L= pU

f

icos Pds* -s f-s D , = pi [(v+dv) sin p - (w+dw cos p] sr s (2.4.1) (2.4.2) (2.4.8) (2.4.9) As

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For plane hydrofoils the dihedral angle P=O and equations (2.4.8) and (2.4.9) become similar to expressions given by WU (1954) and NISHIYAMA (1965) for hydrofoils operating alone, ie. dw=0.

Following the convention in equation (2.2.15), the self-induced drag of the hydrofoil is made up of three parts:

Di = Dhjt line infllaid+D biplane image+Dwavepart (2.4.10)

where each contribution to the drag stem from the corresponding part of the velocity potential.

BRESLIN (1957) derived the wave drag of the free vortex sheet behind the hydrofoil using a doublet distribution to represent the free vortex sheet and he expressed the induced drag of a

hydrofoil in the following manner:

Di = Dlift.line inffluid+Dwall image+ D wave part Breslin (2.4.11)

In figure 2.4.1 the induced drag due to the free surface are compared when using the last two

terms in equations (2.4.10) and (2.4.11) respectively.

60x103

biplane image

wall image

drag due to free surface, -s-eq. (2.4.10), -o-eq. (2.4.11).

wave part. eq.(2.4.11)

wave part, eq.(2.4. top

2 4 6 8

Fnh

Fig. 2.4.1.

The free surface contribution to the induced drag according to the present method and BRESLIN(1957), ie. the drag due to the last two terms of equations (2.4.10) and (2.4.11). Plane hydrofoil with elliptical spanwise load distribution. Ivs=0.2, Aspect ratio A=12.7.

It is observed from figure 2.4.1 that the drag due to the free surface or the sums of the two last terms in equations (2.4.10) and (2.4.11) are identical.

For comparison the drag in infinite fluid for a plane foil with elliptical spanwise loading is given

by, GLAUERT (1959):

CD.

_{= I}

CL2 7r A (2.4.12)

where A is the aspect ratio.

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2.5 A Numerical Scheme for Solving the Direct and Indirect Problem,,

Allowing for Presence of Front Foils

To calculate the characteristics of a hydrofoil, the discretisized lifting line method presented by

BLACKWELL (1969) for airfoils has been adopted to the case of a hydrofoil. The velocity

potential presented in chapter 2.2 is used to account for the effect of the free surface

Changes in inflow conditions due to the presence of one or several front foils are accounted for.. This method is applicable to both the direct and the indirect problem of the hydrofoil.

The indirect problem is important in hydrofoil design. Cavitation limits the maximum load on

the foil. Figure 4.5.1a and b show the front foil chordwise pressure distribution and 'limiting cavitation conditions on the Boeing Jetfoil, NOREEN (1989).

ANGLE OtATTACII. _..ar

SUM,. 41 000 OEFLECTIC. I STAT" I II II Iin rg .5 1101IN OLSON Olt re716.11011001.I031 0" 3.1 Pig. 2.5.1a.

Front foil chordwise pressure distribution

and section shape, NOREEN ,(1.989),

In the indirect problem the spanwise !toad distribution of the hydrofoil is specified bearing in

mind the danger of cavitation, and he spanwise distribution of angle of attack is calculated.

Normally, an elliptic spanwise load distribution is seeked to minimize induced drag.ln the case of a box-foil however, allowing for some load on the struts increases the effective aspect radii of the hydrofoil and reduces the induced drag even more.

Since lifting line theory is used, the term "angle of attack" above is the sum of the zero lift

angle due to the cambered mean line and the actual angle of attack relative to the inflow of a foil section. To avoid cavitation starting from the leading edge the local angle of attack relative to the inflow should be equal to the ideal angle of attack. The ideal angle attack of a hydrofoil section is typically 0.3 to 0.4 degrees.

In, the direct problem the spanwise distribution of angle of attack phis zero lift angles is given

PPAISVP4 PE. LOWIGh

CPCIPO

IICI

tara SIID.0111 COIIPPColpir

..71.1.

Fig.25.1 b.

Limiting cavitation conditions,

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and the spanwise load distribution is calculated. z, c/4 104.1"--40014 3c/4

fr-4*-4

1.4. Vertical surface y,11 A e Fig. 2.5.2.

Distribution of vortices over a combination of lifting surfaces. BLACKWELL (1969).

The foil is according to well known practice divided into spanwise segments as shown in figure 2.5.2 taken from Blackwell, with a constant circulation along the span concentrated in a lifting

line at the quarter chord. This is often described as a system of horseshoe vortices. Each

horseshoe vortex has its local origo, yr, ), at the quarter chord position and mid-span of the foil segment. At the three quarter chord position and mid-span of each segment there is a control point, At each control point tangential flow is required. The present method is easily generalized into a lifting surface method using more than one horseshoe vortex

31

Horizontal lifting surface

Horizontal lifting

/

surface

(34)

along the chord, but here we have settled for one, i.e. a lifting line approach. This is justified since most hydrofoils have an aspect ratio large enough to be properly described by lifting line

theory.

At each control point, the tangential flow condition is expressed by:

Ay, and Aw, represent lateral and vertical changes in inflow velocity from one or more front

foils at Pi,(x,, yi

zs,). op is the angle of attack of foil segment v and the dihedral angle

as shown in figure 2.5.3 a and b. [up,VV, WV] is the induced velocity from all the foil segments

at Pp(x,,yp, zp). Terms due to the longitudinal self induced velocity component, up, are of

second order and neglected in the following.

4

n

y,z

Fig. 2.5.3a. Cross section of foil segment v

Fig 2.5.3b. Foil segment v projected in the plans The self induced velocities are:

nseg 1 VL, =

E rnFu,

n=1 nseg 1 wv =

E

_nr 47r Tt= 1 17, = 0 (2.5.1)

= [U +

+ Ay,

+ Awid (2.5.1a)

= [sin op, sin cos op] (2.5.1c)

(2.5.2a)

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Where F,,. and F,,, are lateral and vertical induced velocities from foil segment rt with unit ie. F = 1.0[m2/s], constant loading, at the control point of foil segment v multiplied by 47.

and follow from equations (2.3.1a) and (2.3.1b):

a211(P)

3212(P,)

a2I3(P.)

Fv,vn = f [

( + + sua On

allay &lay

aqay

.9211 (p

"

a2i,,( ) 52i3 (pm )

"

) COS ,I3nidS

a(aY

°coy

a(ay

a2.r,(P,)

a2.1.2(Pm) a2.1-3(pi,) .

Fw.vn= fAn[ (

ao, + Do,

+

aria z ) sm On

a2i, (P.)

a2I2(P.)

02 T3(P.)

+(

+

+ )cos13,dels'

a(az

a(a,

acaz

All the terms in equations (2.5.3a) and (2.5.3b) are given in Appendix B. The It terms, ie. the contribution from a lifting line in infinite fluid and its biplane image, we choose to substitute with expressions provided by Blackwell . These expressions are the analytical integrals of the

II terms for the case of a horseshoe vortex or a lifting line with constant circulation, with

general dihedral angle. There are two reasons for doing this. They are faster to compute than performing a numerical integration of the I terms above and the problem described in chapter

2.3 of a singularity when calculating induced velocities in the exact plane of the lifting line disappears. The full expressions for F,,, and F,,,, are given in Appendix C.

If we want to solve the direct problem, the tangential flow condition is expressed at each control point as:

1 "eg

rnFv,sn

cos /3--

E

ry--n1

47r

n=1

U sin am sin 13,, Av, + cos 0,A ty, _(2.5.4)

Equation (2.5.4) is rewritten as a system of linear equations from which the circulation of each foil segment, r, n = 1, nseg, are found with a general Gauss elimination algorithm:

AX = B

(2.5.5a)

sin/3,F4, COS Os Fw,vn

=

,u = 1, nseg,n = 1, nseg

(2.5.5b)

4ir

X, = F

, v = 1, nseg _(2.5.5c)

Bs, = U sin a,

sin /34,v + cos 34,w4, ,ii = 1, nseg

(2.5.5d)

When the indirect problem is to be solved the circulation of all the foil segments are known. The distribution of angle of attack and zero lift angles along the span is determined by rearranging equation (2.5.4). 33 (2.5.3a) (2.5.3b) nseg sin 47r n 1

+

=

(36)

nseg naeg

1

= uasin

(sin 19(J_ E rnFv,.

_47r Au) cos Om(

1E

+

Aw)) (2.5.6)

n=1 rt=1

The total force of the hydrofoil is expressed as:

Flier

FN = pU E rn2s

n=1

and the lift as: _naey

L ['LT 12 rn cos On23n

n=1

The sideforce is given by:

nseg

= pU

E Fn sin On2s,

(2.5.9)

n=1

Sectional force coefficients are given by the relation:

Cf n =(2.5.10)

_-Ere

2 "

of which the sectional lift coefficient is given by:

Cl = Cfn COS

We' define the foil lift coefficient by:

CL=

7iPU2Aproj

in which .4proj is the horizontal projected area of the foil. Using equation (2.4.10) the induced drag is expressed as:

nseg

Di = pE

[04, + Av) On (u),, + Aw)cosj3nj2sn

n=1

Following equation (2.5.12) the foil drag coefficient is defined by:

Di

CD, =

_{r I2}

(2.5.14)

.7.Pu --iproj

Calculation of induced velocities at arbitrary positions in the fluid

For a given load distribution the induced velocities from the foil at arbitariry points in the fluid

is determined in two ways. Either by representing the r distribution of the foil (lifting line),

rn, n = 1, rtseg, as a function of span coordinate and use this in equations (2.3.1a and b) or

by direct use of equations (2.5.2a and b), replacing the control points of the foil by the arbitrary points in the fluid.

When studying hydrofoil interaction, one is interested in determining the induced flow due to the front foil(s) at the control points of the rear foil.

(2.5.7)

(2.5.8)

(2.5.12)

(2.5.13)

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Convergence of numerical method

In the following the convergence of the numerical method is tested on a T-foil and a box-foil.

The numerical method is intended as a lifting line method, but shown in figures 2.5.4 and 2.5.5

are also the results of calculations using more than one horsehoe vortex along the chord or

lifting surface theory, although only representing the lifting surfaces as flat plates.

Including the effect of the wave part of the velocity potential in the calculations requires a lot of

computing time. This does impose a restriction when choosing the number of foil segments to

represent the foil. For this reason the load distributions in figures 2.5.4 and 2.5.5 are calculated using only a biplane image to represent the effect of the free surface.

0.12 0.10 0.08 0.06 0.04 0.02

Sectional lift coeff.

Calculated lift and drag, model scale Lifting line, nseg=40

Lifting line, nseg=500

4- Lifting surface, nseg=12x40

Bound circ.r (s ) [rnA2/s]

-1.0 -0.5 00 1.0

SIs

Fig 2.5.4.

Test of convergence of numerical method on a T-foil in model scale using a biplane image to represent the effect of the free surface. Foil characteristics are given in table 2.6.3.

a=0°(50=-1.96°), Fn=6.61, h/s=0.83.

When calculating the box-foil one runs into an additional problem caused by the vertical struts

at the wing tips penetrating the free surface. Letting the struts extend to the mean free surface,

35 CL CD, 0.1136 0.00129 0.1124 0.00129 0.1146 0.00131 0.5

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-z=0, causes the program to crash when including the effect of the wave part of the velocity

potential. Therefore, one can only let the upper end of the struts approach the free surface.

0.4j

c§- 0.3 7-5 0 0.2 (3 "C) a) 0.1 0.0 Vertical struts Horisontal foil

Calculated lift and induced drag, model scale CL Co,

x Lifting line, nseg=4+20+4 0.4049 0.00696

0.01[m] gap between strut tips and free surface Lifting line, nseg=100+300+100 0.4031 0.00695 Lifting surface, nseg=4x(25+75+25) 0.4033 0.00690

S.=-Sh S =St,

if

-1.4 -1.2 1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0. 4 0.6 0.8 1.0 1.2 1.4 S /sh

Fig 2.5.5.

Test of convergence of numerical method on a box-foil in model scale using a biplane image to represent the effect of the free surface. Foil characteristics are given in table 2.6.1.

Fn=4.16, his=0.38.

In figure 2.4.4 the load distribution with the small number of segments is calculated with the

V

S=S

2

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struts cut one centimeter beneath the free surface.

Both the T-foil and the box-foil seem to be satisfactorily represented using lifting line theory and only 30 to 40 foil segments. Leaving a small gap between the mean free surface and the

upper end of the struts on the box-foil has a negligible effect, at least for the box-foil above. A

reduction of foil immersion will increase the effect of the gap. However, a gap of 0.01[m] is

bigger than necessary. The gap does not have to be bigger than 0.001[m] to avoid trouble when running the program.

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2.6 Numerical Examples and Comparisons with Model Tests

2.6.1 Introduction

In the following comparisons between model test results and numerical results based on the method outlined in chapter 2.4, are given. The model tests were performed in the towing tank at MARINTEK with hydrofoils in a tandem configuration to study the influence of distance between front and rear foils at various foil immersions and Froude numbers.

Two foil configurations were tested, of which foil configuration I was identical with the foil system on the foil catamaran shown in figure 1.1, also see figure 2.6.2.

This foil catamaran was intended as a 1160 ton high speed carrier. The foil system of this 90 meter craft consists of four identical hydrofoils carrying equal loads. On the wing tips each foil has a symmetrical non-twisted strut connecting it to the hull. The characteristics of hydrofoils are given in table 2.6.1.

The presence of these struts affect the lift and drag characteristics and such foils are commonly

referred to as box-foils. For strength purposes each foil has a symmetric strut placed at the

mid-span. The effect of this strut on lift and induced drag is minimal and only due to the effect of the thickness of the strut. In the numerical model which is based on lifting line theory this strut is ignored. There is however, an additional interference effect between the boundary layer of the strut and the foil, which may reduce performance. Detailed analysis of this is beyond the scope if this work.

One should note that the box-foils with a relatively high aspect ratio of 8.2, have foil immersion to semispan ratios from = 0.18 to * = 0.38. From the discussion in chapter2.3, see figures 2.3.4 and 2.3.6, one would expect a large influence from the free surface on the flow at the foil itself and also large variations in the flow with downstream distance of each foil caused by the wave part of the velocity potential.

Foil configuration II is similar to the configuration shown in figure 1.2., except that the rear foil struts are placed at the wing tips, see figure 2.6.24. It is typical of the foil systems on two hydrofoil catamarans presently being completed in Norway. These are passenger vessels with displacements between 120 to 200 tons, and design speed between 45 and 48 knots.

This foil system consists of two small aspect ratio front foils with a symmetric strut at the

midsection commonly referred to as T-foils. The rear foil is a box-foil bridging the two catamaran

hulls. The foil immersion to semispan ratio of the front foils in the model tests was A, = 0.83.

Therefore, according the discussion in chapter 2.3, figures 2.3.3. and 2.3.5, the natureof the flow

at the position of the front foils is close to the flow in infinite fluid. The wave part is negligible and the contribution from the biplane image is of the order 10% of the infinitefluid contribution.

Downstream of the front foils there is much less variation with downstream distance than in the case of foil configuration I. This means that the lift and drag characteristics of the rear foil should be fairly constant with respect to distance behind the two front foils.

An important factor is that the free vortex sheets from the front foils in foil configuration II will tend to roll up and translate downward before passing the rear foil, due to the low aspect ratio of the front foils and the long distance between front and rear foils relative to the span of the front

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foils. In this chapter the calculations of the rear foil lift and drag are based on the assumption that the free vortex sheets remain constant.

2.6.2 Test Set Up and Procedure

The lift and drag characteristics of the hydrofoils were measured by mounting the foil to a device with two dynamometers. One dynamometer was used to record foil lift and the other to record foil drag.

The tandem tests were performed by keeping the rear foil set at a constant angle of attack,

(aRF = 2°), and recording the rear foil lift and drag while varying the distance between front and rear foils. At each distance the front foil(s) were set at two angles of attack , (a pp= 0° and 40), to be certain of the effect of increased loading on the front foils on the lift and drag characteristics of the rear foil. Correct angle of attack of the front foil(s) was ensured through the use of high precision level with a scale for elevation.

2.6.3 Foil Configuration I; Two Box-Foils in Tandem

Foil configuration I is shown in figure 2.6.2. The distances between front and rear foil were

chosen to correspond with the foil positions on the hydrofoil catamaran, see figure 1.1, plus an additional position between foil 1 and foil 2 to see the effect of the front foil at a rear foil a

short distance downstream. In table 2.6.1 the foil characteristics are given. The foil sections are

constant along the span. The camber used give a theoretical zero lift angle of 2.90°. Details

of the test conditions are given imitable 2.6.2.

The foils were tested at two immersions, = 0.18 and = 0.38, with the front and rear foils equally immersed. In addition a set of tests were performed with the front foil less immersed than the rear foil. The model tests were performed at two speeds of advance corresponding to 30 and 45 knots in full scale. 30 knots was considered to be close to a take off speed and 45 knots was the design speed. Froude numbers based on foil immersion hence become 2.78 and 4.16 at 0.38 and 4.02 and 6.02 at = 0.18.

In the numerical calculations the wave part of the velocity potential, ie. the I2- and 13-terms in equations (2.5.3a) and (2.5.3b), requires extensive computing time. This restricts the choice as to the number of foil segments used to represent the foils. The number of segments used for the box-foils are twenty on the foil and two or four on each strut at low and deep immersion, respectively, see the discussion at the end of chapter 2.5 and figure 2.5.5.

The influence of the different parts of the velocity potential on the distribution of bound circu-lation for a single foil of this type is shown in figure 2.6.3a and b for the two foil immersions

and the two speeds of advance. The bound circulation is calculated along the struts and the

horizontal foil at the four Froude numbers, Fnh, under three different conditions. The first is in infinite fluid and in the others the effect of free surface is represented by first only including a biplane image of the foil which is correct in the infinite Froude number case, and at last also the wave part, ie. the total velocity potential.

It is clearly seen that the effect of the free surface is important for the lift and drag characteristics

39

*

(42)

of the box-foils. Also, ills seen that the free surface can not be represented by a biplane image alone. The wave part must also be included, although the effect of the wave part reduces with increasing forward speed.

Figure 2.6.4a shows measured and calculated lift coefficients versus angle of attack for a single

foil at s* = 0.38. Projected area ie. horizontal foil span times chord, is reference area for lift

and drag coefficients. Calculated and experimental lift gradients are nearly identical. However, there is a discrepancy in zero lift angle. The experimental zero lift angle is approximately 3.5°

at Frii, = 2.78 and probably 3.8° for Frih = 4.16, whereas the theoretical zero lift angle is

2.90°. This is possibly due to Reynold's number effects and insufficient turbulent stimulation. The two Froude numbers correspond to Reynolds numbers based on the chord lentgh of 3.0 -105 and 4.5 -105 respectively, which is too low to be certain about the experimental results, see BSR Report No.231 (1958).

However, an analysis on hydrofoil sections with arbitrary camber and thickness presented by NISHIYAMA (1958) indicate that there is a shift in the zero lift angle due to the free surface. This may explain our experimental results. Figure 2.6.1 shows the lift gradient and the zero lift angles of a NACA 4412 section for various immersions and forward speeds expressed by the product koh. In figure 2.6.4a koli = 0.06.

-5° U 5° 10°

Fig. 2.6.1. Lift on a NACA 4412 foil section as a function of angle of attack. Dotted line shows the corresponding value in an infinite fluid. NISHIYAMA (1958).

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Fig. 2.62.

Foil configuration one: showing the distances between front and rear foils in tandem foil tests in towing tank.

Distances are given for model scale. k.1:20.

Table 2.6:1

I Foil Characteristics

I Foil:

Thickness distribution .NACA 16-0075 Camber NAGA a=0.8 mod, 1/c-0.025

Semispani Sh [m]

Chord c=0.100 [m]

Struts:

No of per toil 3'

' Thickness distribution NAGA 16-0101

Chord c-0.100 Emil

iTable 2.6.2

Model tect Conditions

Foil Immersion h = 0075.tm] and 01675 Imt hish.0.18 and 038

Dist. Front-Rear Foil 0.6,1.2,2.4 and 3.6 [m] Forward Speed 3.45 and 5.167[m/s],

corresponding to 30 and 45 knots at full scale.

Froude numbers based on depth; 2.78 and 4.16 for h-0.1575 (ml 4.02 and 6.02 for h=0.075 [m] 0.81 [m] 1 1 0.60tml, xish =11-47

it-I Foil 1, Front Foil

q 11 I, II II 2.40' Vol, xish =5.901 II I V ri I/ 1 .3.60 IMLX/Shi= 8.85 Foil 2 '11 Foil 3 11 1, .1 11 11 Foil' 4 =0.407 41 I I I

II

I I I 1.20 [m], = 2.95

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0.20 0 8 0.15 0.10 cn 0.05 0.20 = 02 0.15 7:1 0 0S) 0.10 0.05 Fig. 2.6.3a.

Calculated effect of free surface on load distribution along span and struts. a=0°(J30_=-2.90°), h/sh=0.38.

0.25

...

infinite fluid

---- inf. fluid and biplane image tot. vol. pot., Fnh=4.02 tot. vol. pot., Fnh=6.02

infinite fluid

---- int fluid and biplane image tot. vol. pot., Fnh=4.16 tot. vel. pot.. Fnh=2.78

Fig. 2.6.3b.

Calculated effect of free surface on load distribution along span and struts. a=0°(30_=-2.90°), h/sh=0.18. I I I i I -1.0 -0.5 0.0 0.5 1.0 S./St, i -1.0 I -0.5 I 0.0 S /sh 1 0.5 1 1.0 0.25 ---0

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A plane elliptically loaded foil with the same aspect ratio in infinite fluid has the lift gradient, 'GLAUERT (1959), ie. an increase in CL of 0.088 per degree. This compares

well with our box-foil, which is to be expected since the negative effect of the free surface on lift is to an extent opposed by the positive effect of the struts on the lift.

Lift versus drag coefficients at = 0.38 are shown in figure 2.6.4b. Only induced drag is

calculated, whereas the experimental drag is the sum of induced drag and profile drag.

The profile drag can be estimated from the following expression, SCHLICHTING and TRUCK-ENBRODT (1967):

CDpe"ii,

2CF(1 + b)

(2.6.1)) whith b= 2 to 2.5. CF is the friction coefficient for a flat plate. From an engineering point of view the ITTC (1957) correlation line is a good approximation for CF.:

CF 0.075 (2.6.2)

(log Rn 2)2

Applying equation (2.6.1) over the total wetted surface of a box-foil and using b 2, the profile drag coefficient based on the horizontal projected area of the box-foil become 0.026 and 0.024 at Fnh equal to 2.78 and 4.16 respectively. This is in reasonable agreement with the gap between experimental total drag and calculated induced drag as seen in figure 2.6.4b. Obviously, due to the difference between experimental and theoretical zero lift angles there will be a small difference between the experimental and calculated induced drag. Using figures 2.6.4a and 2.6.4b for CL = 0.4 gives a difference in induced drag coefficient of about 0.002.

The effect of foil interaction is shown in figures 2.6.6 through 2.6.23. In figures 2.6.6 to 2.6.11 the load distribution on the rear foil is calculated when the foil is operating alone and behind

a similar front foil. This is done for all test conditions. It is seen that when the rear foil operates close behind the front foil, at Lc = 6 and. 12,the downwash from the front foil results in a reduction of the bound circulation on the horizontal part of the foil regardless of foil immersion and forward speed. At positions farther behind the front foil, at = 24 and Lc = 36, forward speed and immersion (of the front foil) decide whether the rear foil operates in an area of downwash or upwash.

At the struts however, the amount of sidewash determines the bound circulation. Not all the

sidewash on the struts is due to the flow behind the front foil, some of it is self induced. It is seen from figures 2.6.6 to 2.6.9 that the loading on the rear foil struts generally increase as a result of the flow from the front foil, ie. the sidewash on the rear foil struts is directed inward to the midspan. The exception from this is when parts of the rear foil struts are deeper than the front foil as in figures 2.6.10 and 2.6.11. The loading on the struts result in a sideforce which due to symmetry is equal to zero. However, as indicated in figure 2.6.5 there is a forward thrust on each of the two struts which helps reduce the induced foil drag.

43

=

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0.5 0.4 0.3 0.2 0.1 0.0 1 CD,,,,=2.CF-(1+21), see eq. (2.6.1) 4.

+ Fnh=4.16, exp. total drag x Fn1,=2.78, exp. total drag

Fnh=4.16, calc. induced drag Fnh-2.78, calc. induced drag

I I i I I

0 5 10 15 20

CD

Fig. 2.6.4b. Lift versus drag.

Fnh 4.16, experiment Fnh = 2.78, experiment Fnh = 4.16, calculated Fnh = 2.78, calculated 25 30 35x10-3 -4 -3 -2 -1 2 cc [deg]

Fig. 2.6.4a. Lift versus angle of attack. h/s=0.38, 130=-2.90°.

0.5 0.4 0.3 0.2 0.1 0.0 + ,

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Resulting perpendicular

inflowat strut section

Thrust Section krce

_40-1 Sideforce

Fig. 2.6.5. Resulting thrust from sidewash on vertical struts.

Figures 2.6.12 to 2.6.23 give experimental rear foil lift and drag at the different positions behind the front foil at the various immersions and forward speeds. Also given is calculated lift and induced drag. The shift between experimental and calculated lift is caused by the difference between the experimental and theoretical zero lift angles mentioned above, see figure 2.6.4a. Apart from this, the experimental and calculated rear foil lift are almost identical throughout all the test conditions.

Ignoring the effect of inflow angle on the profile drag, the changes in the rear foil drag when operating behind the front foil can be regarded solely as changes in the induced drag. In support of this is the almost constant difference between experimental drag and calculated induced drag which we therefore, like above, attribute the profile drag, see figure 2.6.4b.

The close agreement between theory and experiments concerning the changes of the lift and drag characteristics of the rear foil at the various positions behind the front foil is explained by the high aspect ratio of the box-foils and the fact that the distance between front and rear foils at the most are equal to only 4.4 times the foil span. The result of this is that the free vortex sheet

of the front is not likely to be strongly rolled up at the position of the rear foil. The applied

theory assumes the free vortex sheet to be constant as discussed in chapter 2.2.

When the rear foil operates in a in a upwash both experiments and calculations show a reduction of the foil drag and when in a strong downwash the rear foil lift is reduced and the drag increased. When the vertical induced velocities from the front foil are close to zero or moderate the lift of rear foil is hardly affected whereas the sidewash on its struts still help to reduce the drag.

too

45

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-The model tests and calculations for foil configuration I show that it is possible to design a foil system for a hydrofoil catamaran with less total resistance than the sum of each foil neglecting the interaction, if the spacing between the foils is properly taken care of. Here, both experiments and calculations show that the greatest benefit from the front foil is at the position of the third foil at 30 knots and the position of the last foil at 45 knots.

Given an angle of attack of 2° of both front and rear foils, the rear foil will at the optimum

position have a lift increase of 10 to 15 % and a similar reduction of drag. Assuming a constant rear foil lift the drag reduction would be even bigger.

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RE alone x/sh=1.47 ---x/sh=2.95 - - -x/sh=5.90 --- x/sh=8.85 -1.0 -0.5 0.0 s./sh 47 Fig. 2.6.6.

Calculated load distribution along the rear foil span and struts when operating alone and behind a similar front foil.

cy.F =2° & cc,,,E =2° (flow 2.90°), h/s=0.18, Fnh=6.02.

-RF alone x/sh=1.47 ----x/sh=2.95 - - x/sh=5.90 --x/sh=8.85 0.5 Fig. 2.6.7.

aFF .2° & ariE =2° (Poo° =-2.90°), h/s=0.18, Fnh=4.02. 0.40

...

0.35 0.30 ,E c.) 0.25 ... 0.20 ... / 0.15 0.10 -1.0 -0.5 0.0 s./sh 0.5 1.0

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RE alone x/sn=1.47 --- x/sh-2.95 - - x/sh=5.90 --- x/sh=8.85 Fig. 2.6.8.

CCFF =2° & =2° (130.. =-2.90°), h/s=0.38, Fnh=4.16. 0.4 0.3

...

... ... RF alone x/sh=1.47 ----x/sh=2.95 - -x/sh=5.90 ----x/sh=8.85 s'ish s./sh Fig. 2.6.9.

a.FF =2° & aRF =2° 2.90°), h/s=0.38, Fnh=2.78.

-1.0 -0.5 0.0 0.5 1.0

0.5 1.0

-1.0 -0.5 0.0

0.4

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0.2 0.4 0.3 0.1 0.0 0.4 0.3 0.2 -1.0

...

-az S IS Fig. 2.6.10.

an: =2° & afiF =2° (pe,.., =-2.900), h/sFF=0.18, h/sRF=0.38, FnhRF=4.16. 49 1 0.0 s./sh RE alone ---x/sh=1.47 ----x/sh=2.95 - -x/sh=5.90 - -x/sh=8.85 RF alone x/sh=1 AT x/sh=2.95 - - x/sh=5.90 -- x/sh=8.85 0.5 1.0 / Fig. 2.6.11.

ccrF =2° & Cf/RF =2° (50, =-2.90°), h/SFF=0.18, h/sRF=0.38, FnhRF=2.78.

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0.40 0.35 0.30 -(54 0.25 0.20 0.15 25x10.3 20 15 c.) 10 5 --b-- a,FF=0°,calc. CyF=0°,exp. aFF-e,calc. aw..4°,exp. RF alone. cab. RF alone, exp. 1 2 4 6 8 xish Fig. 2.6.12.

Experimental and calculated rear foil lift versus distance behind front foil. cz.RF=2°, Fnh=6.02, h/sh=0.18, Door,=-2.90°. ... ... . 1 I i 2 4 6 8 x/Sh Fig. 2.6.13.

Experimental rear foil total drag and calculated induced drag versus distance behind front foil. aw=2°, Fnh=6.02, h/sh=0.18, u3000=-2.900.