Regression analysis of surface piercing propeller series

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TiCHNISCHE UNiVERSITEIT Laboratorium voor ScheepshydromectianIca Archief

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REGRESSION ANALYSIS OF

SURFACE PIERCING PROPELLER

SERIES

by

D. Radojcic., D. Matic.

ABSTRACT

Surface piercing propellers, together with fully submerged supercavitating propellers

and

waterjets, are the only propulsors which can operate efficiently at very high speeds (above 40

knots). To facilitate computer application for feasibility studies and

preliminary design phases,

adequate mathematical models are

needed. Consequently, this paper is directed

towards

mathematical representation of limited

methodical series of four bladed surface piercing propellers

-Rolla Series - published previously by Rose and Kruppa 1991 and 1993. Series

consists of five

4-bladed propellers (P/D = 0.9, 1.1, 1.2, 1.4 and 1.6) all having blade-area-ratio

(k/A0) of 0.8. The

tested immersion ratios (h/D) were

30.0, 47.4 and 58.0 %, cavitation numbers (a) correspond to

atmospheric pressure, 0.5 and 0.2. Mathematical models are obtained through regression analysis of

test data. Three sets of

polynomials are derived for thrust and torque

coefficients - Kt_ KQ = f(h/D,

P/D, .1) - each for one cavitation number. Mathematical models enable computer aided

selection of

surface piercing propellers, replacing tedious

"manual" and sometimes empirical, propeller sizing.

Dept of Naval Architecture

Faculty of Mechanical Engineering University of Belgrade

27 marta 80

11000 Belgrade - Yugoslavia

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INTRODUCTION

C3

asu :

Feasibility studies concerning different propulsor types are of primary importance for successful high speed craft projects. To facilitate this, computer simulation is sine qua non. This

means that hydrodynamic characteristics of different propulsors should be given in the form which is, on one hand programmable, and which, on the other hand, may be used with existing computer routines. Mathematical models obtained through regression analysis of test data are widely used and are proven to be reliable, particularly during (for high speed craft very important) preliminary design

phases.

Consequently, the paper is

directed towards mathematical representation of limited

methodical series of four bladed Surface Piercing Propellers (SPP), Rolla Series Polynomials derived to describe the thrust and torque coefficients enable computer aided selection of SPP,

replacing tedious "manual", and sometimes empirical, propellersizing.

OVERVIEW OF SURFACE PIERCING TECHNOLOGY

Fully submerged supercavitating propellers and waterjets, together with partly submerged

propellers or Surface Piercing Propellers - SPP - are the

only propulsors which can operate

efficiently at very high speeds (above 40 kn), see for example Kruppa (1990) and Allison (1978) The main obstacle for high speed operation - cavitation - is substantially reduced by ventilating

suction side of SPP, while pressure side remains fully wetted Furthermore, SPP enable reduction or complete avoidance of appendage resistance since the appendages are above the water level Low draft requirement of SPP is another attribute that should be pointed out. In addition, the simple and

relatively inexpensive installation may be the reason for their recent popularity. As expected, there are some disadvantages associated with SPP

Relatively high vibration and strength problems may arise from the dynamic blade loads; see Olofsson (1993).

Torque absorption and force generation depends on propeller immersion and shaft angle (as well as vessel's trim), so that inexperienced driver might notbe able

to fully utilize the SPP benefits

Poor reversing and low speed characteristics

Generation of side force, depending mainly on the immersion ratio, dictates the need for SPP to be installed in pairs, in order to eliminate the effects on the

hull

Relatively high vertical force, depending mainly on shaft inclination, is generated as well Note however that, if properly taken into the account, this force might be considered beneficial, as it reduce the downward forces and trim. From the strength point of view, both vertical and side force (neither of which do exist in the usual installations) should be taken into the consideration

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-There are two main concepts of SPP application:

Propeller shaft is mounted to the transom. In this case the screwshaft may be either steerable and trimmable (see Hood and Railton - 1989), orfixed. For fixed screwshafts conventional rudders are necessary for steering. Steerabie and

trimmable devices, although somewhat cumbersome, do notneed rudders and

also allow control ofpropeller immersion

Propeller is placed under the hull in a tunnel which should beventilated; see for example Van Tassel (1989). If installed in a tunnel, effectof immersion change can be obtained by lowering or rising of a control plate (flap) mounted in front

of

the SPP

Immersion change enable full torque absorption, so that, in this respect, fixed pitch SPP

behave as controllable pitch propellers.

Models of SPP may be tested in a towing tank (open-water experiments) or in a

free-surface cavitation tunnel. Testing techniques, as well as some results, analysis and comments

for different propeller types

operating in partially immersed mode are given in Ferrando

and

Scamardella (1996) paper.

AVAILABLE DATA OF

SURFACE PIERCING PROPELLERS

Published data of Surface Piercing Propellers (SPP) are scarce; in the last decade practically there are only the results for Rolla Series presented by Roseand Kruppa 1991 and 1993 Since the aim of this paper is mathematical representation of the test data, in depth information on the test

set-up, test parameters and procedures, etc. can befound in the original papers of Rose and Kruppa. Rolla Series consists-of five 4-bladed propellers (P/D = 0.9, 1.1, 1.2, 1.4 and 1.6) all having blade-area-ratio (A.://k,) of 0.8. The tested immersion ratios (h/D) were 30.0, 47.4 and 58.0 %, corresponding to shaft inclination of 4, 8 and 12 degrees, respectively. Tests were carried out in a free-surface cavitation tunnel under atmospheric pressure (a=atm), aswell as o=0.5 and a=0.2.

Data for the three geometric configurations investigated were given in the design charts in

the following format:

J, = f(1(Q/J5)

for a=0.5 and a=0.2

J, = f(Kr/J2) for cy=atm.

In order to enable the use ofalready existing computer routines, these data were scanned, transformed to numerical form using TechDig program, and than recalculated to another (usual)

format; i.e.

from

=

g/(05)

to

Kt)--f(J), n=f(J) and Kr=f0)

from J. t

=

ITKT/J2) to n---ftJ) and K()=f(J)

Finally, three sets of Kr, K0 = ftJ) data were formed consisting of 716, 490 and 359 points

for a = atm, 0.5 and 0.2, respectively.

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ACCEPTED APPROACH

The approach in this paper is essentially the same as, in that given in Radojcic (1988). Initial mathematical models, each for different cavitation 'number, had the following

form

Kr = C (11/D)K

'(13/DY (J) and K0 = Z CQ (h/D) (P/D)Y (J)1

wherex,y,z.=0. After the 'elimination of negligible terms several models were derived, alt with

Very high coefficient of determination, low standard error and high F-value. All the above statistical calculations were done with program Statistica 5.0 (StatSoft Inc.).

Independent polynomials were developed for 'KT and KQ data, so that if adequate measures

were not taken, relatively large inaccuracies in

ri would

occur. In order to overcome this weakness,

a pair of

KT

and Ko

models which gave, the best representation for 1=(.1/270(KT/K0) was chosen

from several derived models (for each a approximately 10 for KT

and KO. From the point of

representing only the or only the KQ data, this final model based on best-looking n curves,

!however,,, was not necessarily the best one.

MATHEMATICAL MODEL

Regression, coefficients and polynomial terms of the final KT and K0 pair for o- = atm, 0.5

and 0.2, as well as boundaries of applicability and control information showing average accuracy

(quality) of derived polynomials, are given in the Tables 1 to3..

KT= C, 12I -f-)7Jz .029476 1.86733' -0.02543 0.5412 -0.90944 II0,09939 0.29816 -1.02612 06829* 0.16506 0.01739 _{_} Cf = 03 5 - 5 0.58 '0.3 5-h 038 Ko= f-h-I .11174z \D/ D/ 1

Table - Coefficients and terms of Ktand KO polynomials, boundaries of applicability

- and control information for a=atm.

If Co x Y 0 0 0 0.004422 0 Cic 0 '0 0 -0.009327 1 2 10 1 10 -0.0436161 1 2 3 1 3 -0.230349 ,2 3 1. 0 o

ii

-0.211791 7' 0 0 1 0 0. 0.049428 HY 0 6 -3 s' 0 0.42686 I -3 1 1 0 0 -0.049563 4 A S 3 Z 0 -0.004534 0 4 Co 0 2 11! 3 '3 ; -0.027097 -0.016 0, 0 1 4 7 0 3 31 0.203726, 3 0 I 1 6i '3 -0.148662 3 A 2 ; 2 01160668 3 .3 1 ;0 a 1 -0.023257 0 A 5 3 0 il I 2 0.025579 0.04747 '7, r 3 '.1.11 5 3 2 For 2 -01006865 I 4, I '0 '3 2. 0054768' 2, 1 3 11' .3 3 01004981 0 _IL s 0.9 5 iCe

- 1.6

0.1071a D+0.05-1+ 0.072 ,0.9 h 1K92 0.054-_D+ 0.025. 3 - 0.040 J0.5 J2 03 0.0028 - s - 0.0006 F - 3866 F- 7304 _ = 0.991j R2= 0.995 C, -0.14357 0.73114 -0.29121' -11.21058 4.03713 0.57767 031,104 0.13634 -0.51693 1,..7. I -C 3 I 0 I 0 2 1 =atm 5 1.6

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Fpnm MncTHcVT

71-7 4.7f'i.-

TPT BEOGRAD

pum

ri kin .

9P1 1

M ISIIRCrO 7M

1c4r, ln.in-m Pi

-0.5

0.3 0.58

0.9

- <1.6

IV Sytripowm on High Speed Marine Vehicles Sorrento 18. - 21. March 1997.

REGRESSION .ANALYSIS OF

SURFACE PIERCING PROPELLER SERIES

by

fl Radojck, 1). Matic

z: 0 5 0.3 0_78 0.9 ..;;; ---- <1.6 h KT

0.036-

0.025.

J - 0 021 ic

lt

0.03-1-1±0.008 - J - 0.012 D -Q D h

Kr

:r.:'0.179--h-4-03-J - 0 224

r

0 054-+0.1.-.1 - 0.081

1) `Q D s =0.00292 S 0.00073

F 3470

F = 6696

R2-0.9921

R2 0 9941

Correctcd coefficients and terms of Kt and q pq/vriornilds,

boniridarie: ofapplicability and control infoiquation for

-

0.5

-LiI eA,

_s

g. Toy

K. =

CT .1 .±!-. I

.1 1-)

D

h"

KQ=1"JCc"I' ID)

P )7

C-r

C

x

y

0.0162.! 0 0 4.11111.1.11 0 0 004379 0 0 -1,35691 7 1 0

-026266

3 0 3 -6.45123 2 0 2 0.006592 0 3 0 5.39744 2 1 2 -0.005377 0 0 3 0.57133 3 3 0 0.0355 3 3 0 -0.50995 1 3 0 0.001067 0 0 6 2.75216 2 0 1 0 503131 3 1 2 0.2329 1 4 0 -0.07966 3 3 1 0.56176 1 2 0 0.106792 4 0 0

13992

3 0 1 -0.001416 0 4 1 -2,27151

-

1 ₁ ₃ -0.265213 4 1 1 .0.937>0 .,. ' 1 1 -0.002853 1 1 2 -144V13 4 i 1 -0,01404-5 2 0 3 -0.24139 1 2 2 0 18728 4 '3 3 0.00788 0 1 4 2.0235 1 0 3 0.04242

'

4 3 = y. 0

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K, 2 0.036---0.025-J -0.021 K, 2 0.03- 4-0.008-3 - 0.012

KT 5 0.179-h4- 03.3 - 0.224 K 5 0.054-4- 01- - 0.081

4= 0.00292 s = 00073

F = 3470 F 6696

= 0.9921 = 0.9941

Table 2 -

Coefficients and terms of K, and lc? polynomials, boundaries of

applicability and control information for o---0.5.

(ti (1-P5Y ,17

Table 3 - Coefficients and terms of K,

and Ko polynomials, boundaries of

applicability

and control information for a=0.2.

K = (i-P3) K'4C° (11-'5/ (Ii./ CT x y Co Y z 0 01621 0 0 0 000-4570 0 0 0 -135691 2 1 0 -0262866 3 0 3 .640123 2 0 2 005)6592 0 3 0 559744 2 1 2 -0001377 0 0 3 0 57133 3 3 0 00353 3 3 0 -050905 1 3 0 000)067 0 0 6 277216 2 0 1 0 503131 3 1 2 0 2329 1 4 0 -07966 3 3 1 056)76 1 2 0 0)06702 4 0 0 1 38992 3 0 1 -0001416 0 4 I -227151 3 I 3 -02652!) 4 1 1 -095285 2 3 1 -0002853 1 3 2 -144273 4 1 I -0.014045 2 0 3 -024139 I 2 2 01812! 4 3 3 30078! 0 I 4 2.0135 3 0 3 0 04242 3 4 4 KT = C, .

(t)

(-P6-) = C cT 0I734 0 0 0 0 04658 0 0 0 0.2819 I I 0 0 70708 3 2 0 -2.3714 0 2 0.26904 0 0 2 0 5602 0 0 3 -015156 0 0, 1 3.5517 3 2 0 0673)3 2 0 2 80275 2 1 I 00145) 1 0 0 -0.3251 0 3 2 0.65194 2 I 1 -101546 3 1 1 -0.7138 I 0 2 -28841 2 2 0 -0.01232 0 3 I 2.6387 3 0 2 -167955 5 1 1 0.105 0 3 3 0 0049 0 4 0 -0.182 0 0 4 -067774 2 2 0 0.0691 0 4 0 0.00248 0 1 5 -0.1768 0 3 0 -005)75 0 0 3 0 1114 3 0 3 0.11697 1 2 0 0.1141 0 2 1 -025)6 0 0 1 0.2096 0- 3 I o = 0.5 = 0.5 03 5 - s 0_58 03 5-h 0.58 0.9 5 - 5 1.6 09 5 -135 1.6 a= 0.2 0.47 - 0.58 0.9 5 -D 5 1.6 K 20025 1 20.7 s =, 0.00247 cr= 0.2 047 -h 0.58 0.9 5 - 5 1.6 I 207 0.0006 K, 2 0.047 15+ 0.017-3- 0.032 F -3794 F 6302 0.99475 R2 = 0.99613 -= -y x y ' -a

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Graphical presentation of the mathematical models are given in the Appendixes I. 2 and 3 for (3=atm, 0.5 and 0.2, respectively. Shown are the values for h/D=0 474 which were employed,

and for h/D=0.55 which were not employed, in the development of the polynomials.

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

Three sets of polynomials were derived, each for one cavitation number. Use of existing computer routines for power prediction, as for example the one presented by Radojcic (1991) is enabled by torque and thrust coefficients given in the form KT, Ic) f(h/D, P/D, J). Furthermore,

proposed mathematical models for SPP Rolla Series are relatively simple, yet the average

discrepancies for n are approximately 1%. The data for (3=0.2 did not allow, however, to obtain

satisfactory equations for h/D<0.47.

In the next research phase a could be treated as an independent (explanatory) variable

together with h/D, P/D and J. This would produce single, but more

complicated mathematical

model

Since SPP are installed in

pairs, estimation of side

force is

not of primary

importance for power predictions for low transom deadrise angles. For higher deadrise

angles, estimation of effective vertical force, which is a function of both vertical and

side force (Rose and Kruppa, 1993), however, is very important, as for outward turning propellers it reduces the trim and therefore influences the resistance. Consequently, as

design optimization must be done on the whole system (hull

and propulsor), the

mathematical representation of effective vertical force for Rolla Series should be done in

the next research phase.

NOM EN (:LATU RE

F-test

h/D - immersion ratio

advance coefficient - V/nD

KT - thrust coefficient based on horizontal thrust - T/pri2D4

K,, - torque coefficient - Q/pn2D5

P/D - pitch ratio

multiple R squared (coefficient of determination)

standard error

horizontal propeller thrust -Faii cosa - F,,, since

propeller efficiency based on horizontal thrust - TV/27nQ =-- (J/27r)(KT/KQ)

- cavitation number based on velocity - 2(p0-pv)/pV2

-T

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-REFERENCES

Allison J. (1978) - "Propellers for High-Performance Craft", il/larine

Technology, Vol. 15, No. 4,

1978.

1:errand° M. and Scamardella A. (1996) -

"Surface Piercing Propellers: Testing Methodologies,

Results,

Analysis and Comments on Open Water Characteristics"Vmall

Craft,

Marine

Resistance d- Propulsion

,Sympo.siumSWAME, Michigan, May 1996.

Hood F. and Rai1ton D. (1989) -

"Arneson Surface Drives", 7hird

Biennial Power Boat

Symposium. SNAME, Miami, Feb. 1989.

Kruppa C. (1990) - "Propulsion Systems for High-Speed Marine

Vehicles", Second Conf on

High-Speed Marine Craft, Kristiansand, Sept. 1990

Olofsson N. (1993) - "A Contribution on

the Performance of a Partially Submerged Propellers",

.

.econd Int. Conf. FAST '93, Yokohama,

Dec. 1993.

Radojcic D (1988) - "Mathematical Model of Segmental Section Propeller

Series for Open-Water

and Cavitating Conditions Applicable in CAD", Propellem '88

,S.,"vnipasiuniWAME, Virginia

Beach, Sept. 1988.

Radojcic D. ( 199 I) - "An Engineering Approach to Predicting the

Hydrodynamic Performance of

Planing Craft Using Computer Techniques",

Transactions TUNA, Vol. 133, London, 199

Rose J

and Kruppa C. (1991) - "Surface

Piercing Propellers, Methodical Series

Model Test

Results", First Int. Conf. FAST

'91, Trondheim, June 1991.

Rose J., Kruppa C. and

Koushan K. (1993)

-

"Surface Piercing Propellers,

Propeller/Hull

Interaction", Second hit. Conf. FAST '93, Yokohama, Dec. 1993

Tassel van G. (1989) - "A

Ventilated Tunnel/Surface Piercing

Propeller System", Intersociety

Advanced Marine Vehicles Conference,

A1AA-89-1538, Arlington, June 1989.

Engineering,

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-Appendix 1 -Ccri-J diagrams of SPP Series- a =aim - according to polynomials given in Table 1

0.2 0.18 0.16 0.14 0'.12

if0.1

la.08 -0.06 0.04 0.02 0

h/D =0.474 P/D =0.9 - 1.6

11111.1

_pH. ...III=

MW-a-gragei4MEVMMMS

www-mmmmmmmmmwmkimmm

.4w wwwm-mmiomMoNalimmmm..

447d. 3:0FAIIIIIMEN1111111i

rAVE'APIPRIMI1111011111

>%AlIMMINESISMOSESEREMEM

IIIHMEEMonle

h/D = 0.55 P/D = 0:9 - 1.6

_MN

111110111111111

SIMONCESNMENEME

mweenemismansta

ANONWilloWIRMININIMIN

eASEPT

ISQ:111111111

PASINESESEMONESNOSEVE

A111111.11111311MILIENEMI

1111111111MLIMM111ALA

T

0/

06

0.5 0.2 0.18 0.16 0:14 0.12 0.1 0.08 0.06 0.04 0.02 0.7

06

05

0.4 Os. 0.3

02

0.1 0 eN1 fn.

0

-0

0

-J -- 0.1

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Appendix 2 - Kricrn-J diagrams of SPP Series - a = 0,5 - according to polynomials given in Table 2

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.18 0.16 0.14 0.12 Cr 0.1 :2. 0.08 0.06 0.04 0.02 0

IM

It:

I 11-

alk.ThA I 1 I

SKAIM111111111111111111111.

WrirtilE111111111111111111111111111111

"-IviLI-4';;_"41111

oo

h/D =0,474 P/D = 0.9 - 1.6

h/D = 0,55 P/D = 0.9 - 1.6

0.7 0.2 0.1 0 1.11 00

"-.dv""11111111111111111111111.111111111.

-:;1211-1.41%,11.4%.,

- 0.5

-r.vial%

wmft.,7141111Mi 61111111111111111111

1111111RM111111611111111

111111511111111:1111111M111

- 0.6

- 0_5

0.4

03

0.7 0.6 0.4 4.) 0.3 0.2 0.1

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Appendix 3 - KT-K,Q-41-.1 diagrams of SPP Series -

= 0.2 - according to polynomials given in Table 3

02

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

h/D = 0.474 P/D = 0.9

- 1.6

0.7

02

0.18 -0.16 0.14 0.12 0.1 0.08 0.06 0.04

0.02

-0

4:4*.-17.11111101K111111EMii.

rellI11111111MINIIM111111112111

T.:

11111111111111111111111111

'4e.M1M11.4161ff.

SLIEMSIEMEIP.11111111111

110101111111113111111MEILIIIIIII

NIIMMIIIMIIMIIKftb."91111111111

IMAM

h/D = 0.55 P/D = 0.9 - 1.6

0.6

. --A0.7421111MillelL11131111111111

0..5

44?:1

11.1111111111M11111111111 0.4

03 IntillEOLNEMMEAMM1111111

-02

0.1 -I

T111

0 00 Crs eS1 00 00. v). 1.4 0.6 0_5 0.4

03

.2

02

0.1 0

d

-