Supercavitating propellers - Lifting-line theory and calculations

by

Poul Andersen

Abstract:

Lifting-line theory for calculating the characteristics of a

supercavitating propeller is derived. The theory makes use of induction factors and is applicable for propellers of normal

supercavitating geometry and for non-zero cavitation numbers.

By comparison with model tests correction factors are developed to improve the numerical results and to correct for the

short-comings of the lifting-line theory.

Results are shown from calculations in which the propeller data are varied systematically thus making a small propeller series.

Introduction.

Since supercavitating propellers and the special theory

for supercavitating profiles first appeared, the use of

these propellers has been rather limited even though much

work has been done to solve their many problems.

The variation of the cavitation number, which has a signi-ficant influence on the successful supercavitating perfor-mance,has brought an extra dimension into the problems which in connection with the special supercavitating profiles,diffi-cult and expensive to manufacture, has made experiments in the cavitation tunnel extremely laborious and costly. So what one needs is a simple and rather accurate theoretical method for predicting the characteristic data, i.e. propeller

thrust and torque, for a supercavitating propeller. It is

important that the propeller characteristic can be deter-mined for a wide range of advance ratio J. This is because

supercavitating propellers are used by small high-speed

crafts such as planing crafts and hydrofoil boats which have

a large 11hump" in the resistancecurve _{at take-off condition,} fig. 1. A propeller which gives the required thrust at

R

take - off

design

I

V

Fig. 1: Typical curve of resistance for a planning and hydrofoil craft. design speed does not automatically give the required thrust during take off and this condition has to be examined

The first theoretical design procedure for supercavi-tating propellers was proposed in 1958 by Tachmindji and

Morgan at NSRDC, who made use of lifting-line theory, [1],

but the predictions did not agree too well with later

expe-rirnents, (2], (3], probably because the influence of an

in-crease in thickness of the profiles from considerations of strength was underestimated. The design procedure developed by Hydronautics appears good as referred to by Barr, [4],

but the details published were very few. The most compre-hensive experimental work has been carried out by Hecker, Shields and McDonald at NSRDC, [5]. Cox [6] used lifting surface theory, taking into account the cavity and profile by means of a source-sink distribution, but his equations

seemed very complex and no numerical result was obtained.

Nishiyama developed in [7] a lifting line theory in which he used Lerbs' induction factors [8] and where the necessary reduction from 3 into 2 dimensions when a blade element is

considered was justified. The numerical result presented

here seemed promising and since the equations are rather

P.n

Fig. 2: Velocity and force diagram for a blade-element

of the propeller a distance r from the axis of

rotation. Angular velocity u = 2irn axial

velo-city (to propeller disc) VA pitch P.

Lifting line equations.

Introducing the circulation r the lift for a blade

element of width dr and a distance r from the axis of

rota-tion, such as the element in fig. 2, is given by Kutta

Jou-kowski law:

dL(r) = PVRr(r)dr = ½pVc(r)cL(r)dr (1)

where is the resulting incident velocity, c is the chord length and _CL is the lift coefficient.

This can be written in the form

CL(r) = f(c,r)c(r) + cLO(cJ,r) (2)

where is the angle of incidence, a is the local cavitation number.

Inserting (2) in (1), we obtain

f(r) _{= ½VR c(r) cL(o,r) + ..cx(r)f(a,r)} (3)} The circulation as defined by (1) has to be taken around a line i.e. the lifting line to which the circulation is bound.

/

Free vortices /

S

Fig. 3: Lifting line approach to the propeller blades. Helicoidal trailing vortices.

The circulation around the lifting line of the

pro-peller blade will give free vortices lying in the helicoidal trailing surfaces behind the propeller since a vortex cannot

stop abruptly in the water (Ilelmholz condition). The heli-coidal surfaces are described by the induced hydrodynainic pitch angle B1.

The strength of the free vortices shed over the distance

* *

dr in the position r will be

aF(r) dr*

/

Introducing Lerb's induction factors 4iru'(r) _* 'A * (r-r) r'(r 4rru' (r) t * - * (r-r) r' (r )

where u' and u are the axial and tangential velocities at position r , resulting from the free vortices of

*

strength r

(r*)

shed from the position r . The vortices

bound to the lifting lines of the propeller blades give no

contribution to the induced velocities. The induction

factors are thus purely dependent on the propeller geometry, i.e.

* *

'A =

IA(r,r

,1(r ),Z)

T T

Z being number of blades, and can be computed once and for

all, e.g. the curves of [8].

For moderately loaded propellers the induced velocities are relatively small and the toal induced velocity u,fig.2,

is nearly perpendicular to _VR.. So from fig. 2 we obtain:

UCOS1

+ USifl

VR r X(r) = (6), where

tan1

A (r) r

Also referring to velocities _{VR VG.}

The induced velocities result from (4) and integration

* 'A

ar(r)1

T *

* dr (7)

r-r 4-. ).,...

Making use of (5) ,

(6) and (7)

a1 -

* *

{rIA(r,r )

+ 41TVG

/r2+A2(r)

r =rb

_{* *} 1 1

r(r)

1 * ) *

ar

r-r

R

+ A(r)IT(r,r )}dr

Introducing the propeller pitch, fig. 2:

P(r)

VA

a + a

= Arctan

Arctan

I

2irr

2rvnr

Substituting this and (8) into (3)

r(r) =

VGc(r){cLO(G,r) +

f(,r){Arctan

- Arctan

_2rrnr

VA

R J * 1

r(r)

1 * * * *

{rIA(r,r )+X(r)IT(r,r )}

4TTVG

/r2.A2 (r)

r

_rb

r

r-r

_*

dr }}

(10)

This is an integral equation for determining the circulation

around a blade element of the propeller.

The practical solution of (10)

is obtained using Lerbs'

method [81 by introducing

= r/R

=

(1+r) -

(1-r)cosç

and expanding into a Fourier series

r((p) = 7TDVA

G sin(p)

p=1

-' * * =

(12)

p=0

=

icpcosp*

Inserting this (10) becomes 2 ₂ c(p) G sin(np) = (7Tr') D 2rr { ( 1 \ Arctan

(1

Arctan

1T1

-wr / 1 1 iG {ha(P)

1+()Z(7rr1)2 1-rb

ii=1 M where h((p)

ip)

sinp cosv-'-cosii ht(p)

\)0 jt,)

v=p+l i(c4) U

If the Fourier series involve only N terms the summation

will also have only N elements and (13) - (13a) will be a system of N linear equations which is solved quite

easily for G.

With the Fourier coefficients G known,the thrust and

U torque coefficients Q K T KQ = pn2D5 T _pn2Dk can be computed.

Regarding the forces of fig. 2 one gets (R '7 * Km = {L(r )cos pn2D _* r =rb (14) r*)

_{+ D(r))sin1}dr}

(15) R * * * * * K Z {L(r )sin1+ (Df(r )

+ D(r ))cos1}r dr

pn2D5 r =rb + x

T)

₍₁₃₎ + ht(tp)}}} (13a)

The friction and drag forces are given by coefficients

D _{= CD}

pVc

and (15) will finally be

c(r)

Ut

/2

V

where the advance ratio J = is used. nD

The induced velocities are given as

u(p) =

V

A

l-rT

pG h'(tp) b p=l VA pG ht(p) ut(P) _{= 1-rb} = p p

The total and local cavitation numbers _{used here are} p0-e total a

0½v2

p0-e local a--- = ½pV

(Trrl)2

p0 is the pressure at the propeller boss, _{e vapor} pressure. (cD+cf) }r'dr' (17) (18) KT KQ = =

-ZJJ

zj2 1 1J-{( ( 2i

G sinp)(

p=11' G sinp) (1 ii=1 A U D (cD+cf) }dr' (16) 71r'\2c(r) + ) + VA

J) D

KTDKT

const. = const.(1 + 5K

TD

from [9] will normally give fast convergence.

Remembering the expressions for r (1) and (12) the

(21.)

If _KT and K0 for a given propeller at a given J and are to be computed one proceeds iteratively. First the assumption

= is made, (13) is solved, and a new is found which is used in the next step of the iteration which goes on until the required accuracy of is reached. The convergence is usually satisfying. At last _KT and K0

are computed.

Very often the task is to design a propeller which gives maximum efficiency under certain conditions e.g. KT, J and

o specified. Under these circumstances a condition by Betz

as quoted by Lerbs [8] gives

const.

tan1

_{= rt}

By use of fig. 2 and (17)

S1fl 1

pG

-- cos1 l--r

Once more we have a system of N linear equations in _G if only a finite number of terms in the expansion is used.

The first step in this iterative propeller design is to esti-mate the value of the const. (e.g. so that = at r' =

then compute from (19) and solve (20) for G. _KT is found by use of (16). Since the form of the profiles is not

known as yet one can as well use _CD = cf = 0. If the

cal-culated value of KT is not equal to the required one _KTD a new step in the iteration is made. The new value of const, used

required sectional lift coefficient can be expressed as

JD cos1

CL _r'c

Now for a specified chord and angle of incidence the camber and pitch which fulfil the requirement for the lift coefficient

can be found for each blade element and so the whole

characte-ristic for the propeller where cD and Cf are taken properly into account can be made by the method stated above.

As given in the title of {8} this theory is only valid

for lightly and moderately loaded propellers, i.e. for small induced velocities u and u . This condition is assumed

a t

to be fulfilled at least in the normal operating range where

the angle a is small. The presence of the cavities is taken into account in other ways as stated below.

CL and CD for the blade element.

The factors f _{and CLO} in the lifting line equations forming the lift coefficient (2) are computed by treating

each blade element as a part of a two dimensional cascade, fig. 4. By this method the presence of cavities and the

Fig. 4: Supercavitating cascade.

interaction between the blades can be accounted for.

Relating the geometric proporties of the cascade to those

of the propeller gives

d 2rrr

y + ₌

Hsu has carried out computations [10) for lift and drag

coefficients for cascades of the various supercavitating profiles, i.e. Tulin-profiles, Johnson 3-term arid 5-term profiles, described by Johnson [11). His results are the

simple formulas iT c -i

ic-i

-1

CL= - ct(i + tany) + k(l + - ) L

-2

_(cL/

= (1 + ak

siny) single profile

where k is the design-lift coefficient of the supercavi-(1+ tany)

(24)

(CL!)

_{can be found for the various profile}

single profile types by Johnson [11).

The formulas are valid for c/d < 1, y 750, /c > 3 (2 cavity length) which is normally the case for

supercavi-tating propellers. The condition 2/c > 3 is, however, only fulfilled for rather small cavitation numbers. By linearly extrapolating the results obtained by Nishiyarna [12] for a single supercavitatirig hydrofoil a correction factor is applied

s = 0.203 + 0.762

1 c

tating profile and

a = for Tulin profiles a =

-

5 for 3-term

a = 7 for 5-term

z

resulting in

f(c,r) =

(1 (r)tany(r))1s

(25)

cLO(a,r) = k(r)(l + (r))' (1 + (r)tany(r)) 1s

For a negative angle of incidence a the formulas given above are no longer valid. The flow around the profile is

completely different and pressure side cavitation is likely to occur. The lift coefficient will become negative and the

drag coefficient will increase drastically, fig. 5. The slope of CL for negative a and the size of _cD is

deter-mined empirically using values which give a good shape for

Fig. 5: Lift coefficient as function of angle of incidence the propeller characteristic curves.

The propeller will not work as predicted for a < 0, but

since this only occurs at relatively high J values it is of limited practical interest for the designer.

The friction coefficient is in all cases 0.004 as for

one side of a flat plate with turbulent boundary layer.

In the case of optimizing a propeller the required value

0

The formula is only a rough estimation and certainly not

beyond discussion.

Since good results are the only things that count for the practical designer,a correction factor s is used with

which f _{and CLO} in (25) are multiplied. Die to lack

of personal experiments this factor is developed by

com-paring lifting line calculations using the formulas above

with experiments, [5] and [3]. s has the form

= '\C

0.6 + 0.4)

2

b

(27) design-lift coefficient k is required. This is easily found by use of (24)

Correction factors.

In the derivation of the above formulas several simpli-fications and assumptions have been made. The theory of the lifting line is only valid for moderately loaded propellers.

For a working condition at a low advance ratio J relative

to the pitch ratio P/D the propeller is no longer moderately loaded. The lifting line theory treats the flow in some

aspects as two-dimensional thus neglecting the influence of

blade camber and chord length.

Furthermore the cavity is assumed to cover the whole

suction side of the propeller blade. This is not the case if the angle of incidence a is small,i.e. for the pro-peller J P/D. When a is small the cavity sheet is so

thin that for a normal blade thickness the blade exceeds the

cavity and the propeller is no longer supercavitating. From

the photographs of the supercavitating propellers of [5J this phenomenon is estimated to occur for

P/D

0.25

KT

1OKQ

0.20 0.15 0.10

0.05

0

0

Tulin sections

0.6

0.8

1.O 1.2

1.430

0.6

Computation

0.4

Experiment

0.2

1.0 Ti

0.8

0.6

0.4

0.2

Fig. 7: Comparison between theory and experiment for

propeller NSMB 2901 [3] KQ Computation

Experiment

a0 =1.0

a= t.o

riiiii

11%

-Mii1.

0

_0.56!

0o 0.9 .= 0.5 ..=0A Co

-0

0.6

0.8

1.0 1.2 1.4 ₃

Fig. 6: Comparison between theory and experiment for

propeller NSRDC 3969-3 [5]

3-term sections

1.0 Ti

_K0

KT

0.8

0.5

KT

1OKQ

0.4 0.3

0.2

01

in which

P(r=0.7)/D

_-J

2

_l+O.5c°11

0

Normally b = 1, c = 1 is used and the factor s2 corrects for camber and other 3 dimensional effects at rather small values of J when the propeller is more than moderately loaded.

If (26) is fulfilled and the effect from blade

thick-ness is considered to be serious, b = 0.6, c = 1 is

used, the computer program writes out a warning against bad

cavitation proporties. C

is used.

The author is fully aware of the big danger in devel-oping a correction factor which is only valid for the pro-pellers used for this work. However, the shortcomings of

the lifting-line theory are too obvious to be ignored and with the correction factors applied,the calculations should be useful at least in the preliminary state of design. Fig. 6 and 7 show comparison between theory including correction

factor and experiments.

Systematical calculations.

For the present lifting line theory a computer program

has been written. By use of this a series of computations

has been carried out in which propeller data are varied sy-stematically. The accuracy of the calculations is of course

not better than normal for this theory as can be judged from fig. 6 and 7, but at least the qualitative considerations

should be correct. If C = P/D - J < 0.3 (i

ifc >0

then b = I

2-0.6 if c2 < 0 and c _{= (c + 0.5)/0.8}

Effects examined include cavitation number number of blades pitch design-lift coefficient type of profile

blade area ratio

distribution of chord length

and using the procedure of optimal propeller

angle of incidence

The motherpropeller 2000/3 for which data are shown in fig.8 is almost identical to NSRDC's propeller 3969-3. The

re-suits and the variation of data of the propellers are given

in fig. 9-20. The warning against bad cavitation properties

are given if (26) is fulfilled. Only one parameter at a time

Conclusion.

Prediction of the characteristic of a supercavitating

propeller is not simple and many parameters have a signi-ficant influence on the performance of the propeller. The small propeller series computed here by means of

lifting-line theory is an attempt to estimate the influence of some of these parameters. Others,as for instance rake and skew, are considered to be of minor importance. Even if the series is very limited the practical designer may use it by inter-polating in the diagrams with sufficient care.

Even if the results of the lifting-line theory presented

here compared with experiments are not strikingly good; they

seem to encourage the treating of propellers in this way. The quality of the results must be seen, too, in relation to the invested computational effort. The computer

time is very short (e.g. 10 sec. for a whole propeller cha-racteristic on a large computer) and since the lifting-line equations differ only slightly from those for normal pro-pellers, any lifting-line computer program can easily be modified, for instance by adding a special subroutine, to

include the case of supercavitating propellers, provided the

program makes use of induction factors.

The demand for improved results can be fulfilled by

further development of the correction factor. This can be

done either in the empirical way by systematically changing

the correction factors, computing and comparing with

experi-ments, or in the theoretical way by the lifting-surface

equations given by Cox [6].

With good correction factors the lifting-line theory

should be a fast and practical tool for the propeller de-signer.

Expanded view

0.5

KT

1OKQ

0.4

0.3

0.2 0.1 1.1 1.2 1.3

P(r) m 0

0.1 0.2 k

Fig. 8: Propeller 2000/3.

A/R

1.00.5

--4.

0.5

-

.1-III I

I Bad

cavitation

0.4

properties

Fig. 9: Influence of cavitation

number for propeller 2000/3.

0.5

1OKQ

0.4

0.3

0.2 0.1 0 2 000/ 2000/ 20001 2000/ 4 3 4 3

200/2

11 0.6 0.8 1.0 1.2 0

1.43

1.0 Ti 0.8 0.6 II Bad

cavitation

0.4

properties

0.2

Fig. 10: Influence of number of blades (propellers 2000/2, /3,

/4, /b, b refering to blade number) = 0.5. The

propellers are equal at all points, except for the

0.5

KT

1OKQ

0.4

0.3

0.2

0.1 2010 2030'3

Fig. 11: Influence of pitch. See fig. 12 for variation in pitch.

P(r)m

2030/3 50 1.6 1.4-0.6 1.0-2010/3 10"

0.8

2000/3 1.2-2020/3 50 1.0 1.2

1.4

163

1.0

TI

0.8

0.6

0.4

0.2

1111111 Bad

cavitation

properties

0.8-0.6 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

r'

boss

tip

0.3 0.2 0.1 0 KQ 20O3I 2002/.3 2003 2000/ 0.6 0.8 1.0

Fig. 13: Influence of design-lift coefficient. Variation in

design-lift coefficient is given in fig. 14. It is

interesting to note the relatively small differences

in efficiency. 0.4 -0.3

0.2

0.1 -2003/3 2002/3 2000/3 2001/3 1.2 1.4 0 1.0 11 0.8 0.6

0.4

0.2

0 T ¶ j P 0.3

0.4

0.5

0.6

0.7

0.8

0.9 1.0

r

boss

tip

Fig. 14: Distribution of design-lift coefficient for

various propellers.

11.11111

Bad

cavitation

properties

0.5

KT

1OKQ

0.4

0.3

0.2 0.1 0 K0 KT 5000/ 5000/ 303013 11 5000/ 3000/ 2000! 0.6

08

1.0

12

143

0 1 .0 Ti

0.8

0.6

0.4

0.2

Fig. 15: Influence of profile type, for propellers 2000/3 Tulin profiles, 3000/3 with Johnson 3-term and

5000/3 with Johnson 5-term.

Bad

cavitation

properties

0.5

KT

1OKQ

0.4 0.3 0.2 0.1 2500/ 2400/ 2300 2200!

Fig. 17: E::panded view of various propellers.

1.0 0.8 0.6 0.4 0.2 11

Fig. 16: Influence of expanded area ratio EAR, variation

of which is given in fig. 17.

EAR = 0.285

11111

_III

_II

Iii

_III.

_______iu

200/s 2 400/3 = 0.356 = 0.1.45 = 0.535 = 0.624 0.713 2000/3 230013 2500/3

tilt

I Bad

cavitation

properties

06

0.8 1.0 1.2

1.43

Fig. 18: Influence of distribution of chord length as

given in fig. 19. Note that the propeller

efficiency has hardly changed.

I / /

/

/

//

\

' \

2600/3 2700/3

"\ 2000/s

\2800/3

0.2

Fig. 19: Expanded view of various propellers. _Expanded area ratio is held constant.

2700/3,2800 3

r

4

I

2600/3 K0

V_____

___

27 2000

3!U_____

2800 3fl__

2600 2000 2 700/s 2800/3 KT

-0.6

08

1.0 1.2

14 J

0.6

ii hilt

Bad

cavitation

0.4

properties

KT

10K0

0.4 0.1 0.2 0.3 0 11

0.8

1.0 Ti 0.8 Jzl II I II Bad cavitation

0.6

propertieS Angle of

0.4

incidence 10

-

20 30 0.2 1.4 ₃

Fig. 20: Optimizing propellers for various specified angles of

incidence. Chord distribution specified as for

pro-peller 2000/3. Design point = 0.11 J = 0.908

as indicated by the arrow. That this point is not reached is caused by the fact, that the optimizing is carried out neglecting friction and drag while this is not neglected when the whole characteristic

is computed. G = 0.5.

_

I

0.5

KT

10K0

0.4

0.3

0.2

0.1 0 0.6 0.8 1.0 1.2

Notation

e vapor pressure

G Fourier-coefficients of the circulation p

p pressure

p pressure at height of propeller boss V velocity

p density

r circulation

Propellers D diameter

axial and tangential induction factors

i'

Fourier coefficients of induction factors n number of revolutions

P pitch

R radius (tip) r radius

rb radius of boss

r' radius non dimensionalised r/R

UaIUt axial and tangential induced velocities

Z number of blades

induced inflow angle hydrodynamic pitch angle

hydrodynarnic induced pitch angle

A

=rtan1

Propeller coefficients VA J advance ratio K thrust coefficient T T _pn2D' K torcue coefficient -pn2D5 K n efficiency = 0 cavitation number p-e

½pV Profiles c chord length D drag force Df friction drag profile drag

d distance between blades of cascade

f factor, see cL k design-lift coefficient length of cavity L lift force S area of hydrofoil angle of incidence stagger angle c lift coefficient - - + c L ½pV2S 2 Lo D c drag coefficient ½PV2S Cf friction coefficient cavitation number

References:

[11 Tachmindji, A.J., Morgan, W.B.:

The Design and Estimated Performance of a Series of Supercavitating Propellers.

Proc. of the Second Symposium on Naval Hydrodynamics, Aug. 1958.

[2] yenning, Elias,r., Habermann, William L.:

Supercavitating Propeller Performance. Trans. of SNAME, vol. 70, 1962.

[3) van de Voorde, C.B., Esveldt, J.:

Tunnel Tests on Supercavitating Propellers. International Shipbuilding Progress, vol. 9, Nov. 1962, No. 99.

[4] Barr, Roderick A.:

Supercavitating and Superventilated Propellers. Trans. of SNAME, vol. 78, 1970.

[51 Hecker, Richard, Shields, C., McDonald, N.:

Experimental Performance of a Controlable-Pitch

Supercavitating Propeller. NSRDC-Report 1636, Aug. 1967. Cox, Geoffrey G.:

Supercavitating Propeller Theory - The Derivation

of Induced Velocity.

Seventh Symposium on Naval Hydrodynamics, Aug. 1968. Nishiyama, T.:

Lifting-Line Theory of Supercavitating Propellers at Non-Zero Cavitation Numbers.

Zeitschrift für Angewendete Mathematik und Mechanik, vol. 51, 1971.

[81 Lerbs, H.W.:

Moderately Loaded Propellers with a Finite Number of

Blades and an Arbitrary Distribution of Circulation. Trans. of SNAME, vol. 60, 1952.

[91 Høiby, Ove W.:

Skipshydrodynamikk. Kurs 7, Propeilteori II

Institutt for Skipshydrodynamikk, 1971, Trondheim. [10] Hsu, C.C.:

Some Approximated Results For Supercavitating

Flow Past Cascade of Flat-Plate and Cambered Hydrofoils. To be represented at the 1975 ASME Cavity Flow Symposium.

[ii] Johnson, Virgil E., Jr.:

The Influence of Depth of Submersion, Aspect Ratio and

Thickness on Supercavitating Hydrofoils Operating at

Zero Cavitation Number.

Proc. of the Second Symposium on Naval Hydrodynamics, Aug. 1958.

[12] Nishivama, T.:

Lifting-Line Theory of Supercavitating Hydrofoil of Finite Span.

Zeitschrift für Angewendete Mathematik und Mechanik, vol. 50, 170.