development of marine propellers with better cavitation

48

(Read at the Spring Meeting of The Society of Naval Architects of Japan, May 1988)

Development of Marine Propellers with Better

Cavitation Performance

2 nd Report : Effect of design lift coefficient for

propellers with flat pressure distribution

by Hajime Yamaguchi*, Member Hiroharu Kato*, Member

Akihiro Kamijo** Masatsugu Maeda*

Summary

A new propeller design method to obtain the blade section shapes by prescribing pressure distribution is developed, combining a 2-dimensional foil design theory with a propeller lifting surface one. In this method, new blade sections can be designed in accordance with the va-riation of the section lift coefficient CL in a given wake.

Three propellers with flat pressure distribution which was proved effective to reduce cavity volume and pressure fluctuations in previous papers are newly designed , changing the design CL. A propeller with the lowest design CL has flat pressure distribution in the mean wake. A propeller with the highest design CL has the least face cavitation margin which is almost same as that of a MAU type propeller used for comparison. The remaining one has the middle design CL between those of the above two propellers.

Using these new and MAU type propellers, the effect of the design CL was experimentally investigated. The main results obtained are as follows :

( 1 ) For increasing the propeller open efficiency and reducing the cavity volume and cavi-tation noise, the highest design CL was most effective, showing 4.5% higher efficiency and about 10 dB lower noise in the frequency range of 10 to 100 kHz than the MAU type one . The noise reduction is considered due to the decrease in the cavity volume which is amount of the cavity collapsing in each propeller revolution.

( 2 ) For reducing the pressure fluctuations, the middle design CL was most effective, showing about 0.8 times the fluctuating pressure amplitude due to the MAU type one at both the first and second blade frequencies. This result is considered due to the reduction of cavity volume variation.

Nomenclature

Ac=-Mean cavity area on a blade

CL= Lift coefficient of the blade section

Cp=Pressure coefficient normalized by the in-

flow velocity to the blade section

Cpn=Propeller pressure coefficient

(=(P-P0)1(ƒÏn2D2/2))

C0.7R=Chord length of the blade section at 0. 7 R

position

D=Propeller diameter

J=Propeller advance coefficient KT= Propeller thrust coefficient

KQ=Propeller torque coefficient

Kpz=Nondimensional fluctuating pressure ampli-

tude at z-th order (=ƒ¢Pz/ƒÏn2D2))

n =Number of propeller revolution

P=Static pressure

P0=Static pressure at the propeller shaft

center

r=Propeller radius position

R=Propeller radius

RnD=Reynolds number based on the propeller

diameter (=70210 Rnk=Kempf's Reynolds number

(= C0.7R•ãVA2+(0.7ƒÎnD)2ƒË)

SAC=Standard deviation of cavity area on a blade

VA=Propeller mean advance velocity

w=Wake fraction

(x, y) = Cartesian coordinateα/αs=Air content ratio of the water to the

saturated condition at 1 atm

* Department of Naval Architecture, The Uni-

versity of Tokyo** Graduate School , Department of Naval Archi-

tecture, The University of Tokyo

Development of Marine Propellers with Better Cavitation Performance 49

⊿Pz=Single amplitude of fluctuating pressure at

z-th order

ηo=Propeller open efficiency

ν=Kinematic viscosity of water

ρ=Density of water

σ=Cavitation number normalized by the

inflow velocity to the blade section

σn=Propeller cavitation number (=(po-pv)/

(ρn2D2/2))

φz=Phase delay of the fluctuating pressure at

z-th order from the blade position of Ψ=0°

Ψ=propeller blade angle position

1. Introduction

Propeller designers and researchers have been making many efforts to reduce evil effects of cavi-tation such as pressure fluctuation, noise and ero-sion. It can be said that the improvement of pro-

peller blade contour, i. e. highly skewed propeller1), is a typical example to have succeeded in reducing

particularly pressure fluctuations. This result is obtained by the phase difference in cavitation ge-neration due to its blade contour configuration. On the other hand, improvement of cavitation per-formance can also be derived by altering the blade section shapes which directly affect the pressure distribution on propeller blade. Regarding propel-ler blade sections, MAU21, NACA3), Wageningen-134), SSPA'), SRI-B6), KI371, NTR8) type ones, etc. were experimentally and/or theoretically deve-loped and most of them have been used until now.

However,. they were developed, not necessarily on the basis of deep understanding of the relation among blade section shape, pressure distribution and cavitation performance. It can be said, there-fore, that both developing a method to obtain blade section shapes with prescribed pressure distribution and investigating the above relation are important in order to develop a propeller with better cavitation performance.

In previous papers9),10), the authors proposed a

propeller design procedure of combining Eppler's 2-dimensional foil design theory") with Hanaoka and Koyama's propeller lifting surface one12), show-ing that the flat pressure distribution was remarka-bly effective to reduce cavity volume and pressure fluctuations. However, the effect of the blade sec-tion improvement was not necessarily clear at that time since the open characteristics and the load distribution in radial direction of the new propel-lers were different from those of the original MAU type one because of the incompleteness of the des-ign method. Moreover, although Nakazaki et al. and Sato") also confirmed that the flat pres-sure distribution was effective, it has not been clear at what lift coefficient the flat pressure distribution should be given.

On the basis of the above knowledge, the pur-

poses of the present study are : ( 1 ) To develop a method to obtain propeller

blade section shapes with prescribed pressure dis-tribution,

( 2 ) To design a series of new propellers with different design lift coefficients, defined as the lift coefficient where the pressure distribution becomes flat, and

(3) To investigate the effect of the design lift coefficient on the propeller open characteristics and the cavitation performance by making experiments together with the original MAU type propeller.

2. Propeller Design Method

The present propeller design method is shown in Fig. 1. Although this method is basically same as that described in the previous papers9)10), thrust, torque and their distributions in the radial direction as well as blade contour and blade thickness ratio were kept same as those of the original MAU

propeller in order to clarify the effect of new blade sections. This design method is basically divided into two processes.

Firstly, considering the lift coefficient variations and cavitation numbers at the respective radius

positions of the original propeller, 2-dimensional foils with desirable (flat in this research) pressure distribution are designed by using Eppler's 2-dimen-sional foil design theory"). These foils are called"d

esigned 2-dimensional foils", which are to be

Fig. 1 Flow chart of propeller design

50 Journal of The Society of Naval Architects of Japan, Vol. 163

matched with the equivalent 2-dimensional foils

of the new propeller at the respective radius posi-

tions. However, these foils are not practical since

Eppler's theory gives a foil with no thickness at

the trailing edge. Therefore, only the thickness

distribution after the maximum thickness point was

replaced by a 3 rd order polynomial to have the

same trailing edge thickness as the original pro-

peller, keeping continuity of the foil shape up to the 2 nd derivative. Figure 2 shows an example

of the effect of this modification on the pressure

distribution. This is the result at O. 8 R position

of a propeller, MP 016 to be described in the next

section. The solid and broken lines denote the

pressure distribution and foil shape with and with-out this modification, respectively. Although the

foil shape modification is not so slight, the differ-

ence in the pressure distribution is not significant.

Therefore, the foil modified in this manner is a-

dopted as the "designed 2-dimensional foil".

Secondly, a propeller whose equivalent 2-dimen-sional foils agree with the "designed 2-dimensional foils" is designed using a propeller lifting surface theory developed by Hanaoka and Koyamai2). Af-ter some trials, a method shown in Fig. 3 was adopted in this process15). As shown in Fig. 4, the equivalent 2-dimensional camber lines obtained by this method agree approximately with the designed ones. In developing the method shown in Fig. 3, some iteration procedures with more complicated evaluation functions D(r, x) were tried to obtain more excellent agreement between the equivalent and designed 2-dimensional camber lines. Howev-er, largely wavy geometrical camber lines with

poor continuity in the radial direction were obtain-ed without significant improvement in the degree of the agreement. A reason for this is considered the characteristics of the lifting surface calculation. Since in this lifting surface calculation method the

geometrical camber lines are approximated by 4 th

Fig. 2 Effect of foil shape modification to

thicken the rear part of the foil sec-

tion ; MP016, O. 8R, Design Lift Coeffi-

cient

Fig. 3 Method to obtain new propeller blade sections whose equivalent 2-dimensional camber

lines agree approximately with those of the designed 2-dimensional foils


order polynomials with a least-squares-fitting tech-

nique, the equivalent 2-dimensional camber lines

also become to have the characteristics of the 4 th

order polynomials, which could not express the

designed 2-dimensional camber lines exactly. In

Fig. 5, a comparison between the propeller and the

1) O. 8R 2) O. 5R

designed 2-dimensional foil pressure distributions in the case of MP 016 is shown by the solid and the chain dotted lines. Good agreement is obtained except near the leading edge. It is considered that also the discrepancy near the leading edge is due to the camber line expression in the lifting surface calculation. As shown in Fig. 4, the 4 th order

polynomials probably could not express the relati-vely high inclination of the geometrical and the designed 2-dimensional camber lines near the lead-ing edge (more clearly seen in the case of the thicker section, i. e. at O. 5 R). It is estimated, therefore, that the actual pressure distribution is closer to the chain dotted line.

3. Design of New Propellers

Three flat pressure distribution propellers with

different design lift coefficient were newly designed

by the above-mentioned method in order to investi-

gate the effect of design lift coefficient on the pro-

peller open characteristics and cavitation perfor-mance.

Table 1 shows the principal particulars of the

original MAU type propeller together with those

of the new propellers. The MAU type propeller,

MP 002 has the same principal particulars except

the diameter as that used by Takahashi et al.7).

when they developed a KB type propeller. However,

the trailing edge of the present propeller MP 002

was slightly thickened near the tip, since this

propeller was used for cavitation erosion experiment in the beginning. The readers can see the trailing

edge thickness in the offset tables of the new pro-

pellers shown in the Appendix. The design point was selected as

in the wake distribution16) shown in Fig. 6.

The difference in the design lift coefficient among

the new propellers are illustrated in Fig. 7. A

propeller with the lowest design lift coefficient, MP 016 has flat pressure distribution at the mean

Fig. 4 Degree of the agreement between the

equivalent and designed 2-dimensional

camber lines ; MP016

Fig. 5 Comparison of steady state pressure dis-

tribution in the mean wake ; O. 8R

Table 1 Principal particulars of the model propellers


CLB=CL value which gives flat pressure

distribution on the back surface

CLF = The lowest CL value in the region

of no face cavitation

MP018

MP017

MP018

wake. A propeller with the highest design lift

coefficient, MP 018 has the least face cavitation mar-

gin which is almost same as that of the MAU

type one, MP 002. MP 017 has the middle design

lift coefficient between those of MP 016 and MP018.

These differences in the design lift coefficient can

also be explained with the bucket chart shown

in Fig. 8. The most part of the working range in

the wake is included in the bucket of MP 018,

thereby the least cavitation generation is expected

on this propeller. As the result of the section

shape improvement, the buckets of the new propel-

lers are wider than that of MP 002.

Such pressure distribution as shown in Fig. 9 was

prescribed near the propeller tip. In addition to adopting flat pressure distribution to reduce cavity

volume, attention was paid to the followings, based

on the results of the previous research9).10).

Fig. 6 Wake distribution (SEIUN-MARU's estimated fullscale wake)

Fig. 7 Design concepts of three new propellers

near the tip (r•†0.8R) ; MP 016, MP 017,

and MP 018

Fig. 8 Bucket charts and lift coefficient varia-

tion at O. 8R position of the new and

original MAU type propellers

Fig. 9 Concepts in prescribing flat pressure

distribution


(1) ƒµ= 0•‹( Maximum CI_ )

(2) ƒµ = -135•‹( Minimum )

( 1 ) The difference between flat part and vapour pressure should be increased as much as possible.

( 2 ) The flat pressure should not be connected to the leading edge so as to avoid the generation of a steep negative pressure peak at the leading edge in the case of larger angle of attack.

( 3 ) The pressure recovery region in the rear part of the foil should have at least 35-40% chord

MP002 (MAU) MP016

MP017 MP018

length of the foil. The purpose of the items ( 1 ) and ( 2 ) is to sup-

press the sudden generation and collapse of a large cavity. The purpose of the item ( 3 ) is to prevent the trailing edge separation of the boundary layer.

Comparison of pressure distribution at 0.8 R

position among all the propellers is shown in Figs. 5 and 10. The former is the results of the steady calculation using the mean wake and the latter of the unsteady one. It can be seen that increasing design lift coefficient decreases the pressure differ-ence between the upper and lower surface near the leading edge, while increases in the rear part. This result shows that the increase in the design lift coefficient lessens the angle of attack and rai-ses the camber. On the other hard, the MAU type propeller shows less smooth pressure distribu-tion than the new propellers, denoting a gently-sloping peak near the midchord on the back surface.

The principal particulars and the blade section shapes of the new and MAU type propellers are shown in Table 1 and Fig. 11. The blade sections designed by the present method were named UTOP

(University of Tokyo Propeller). It is seen that the camber rises and the pitch decreases with in-creasing design lift coefficient, as predicted in the comparison of pressure distribution. The length of the pressure recovery region in the rear part of the new propellers were increased towards the boss in order to give high efficiency by suppressing the boundary layer development. As a result, the sec-tion shapes of the new propellers approaches those

Fig. 10 Comparison of unsteady pressure

distribution at 0.8R

Fig. 11 Propeller section shapes


of the MAU type one towards the boss. The ra-dial direction continuity of the new propellers were realized by prescribing the pressure distribution

parameters of the designed 2-dimensional foils con-tinuously as well as by adopting a simple evalua-tion function for the difference between the equiva-lent 2-dimensional and geometrical camber lines

(Fig. 3). The parameters of the designed 2-dimen-sional foils and offset tables of the new propellers are shown in the Appendix.

4. Experimental Results

4. 1 Propeller open characteristics

Propeller open tests were carried out at the Small Towing Tank of Akishima Laboratories (Mitsui Zosen) Inc. The propeller Reynolds num-ber RnD was 6.0 x 105.

Comparison of thrust coefficient KT, torque coef-ficient KQ and propeller open efficiency )20 based on the advance coefficient J is shown in Fig. 12, toge-ther with the results of the lifting surface calcula-tion. The thrust coefficients of the new propellers agree very well with those of the MAU type pro-

peller and the calculation especially at the design point. On the other hand, the torque coefficients of the new propellers are lower than that of the MAU type one, resulting in the open efficiency increase. Figure 13 shows the comparison of the

open efficiency based on Kr/ J2. P. At the design

point, the efficiency of MP 016, MP 017 and MP 018 is respectively 1. 1%, 3.1% and 4. 5% higher in the ratio than that of the MAU type one. It is considered that one of the reasons for this is the reduction of boundary layer development due to the smoother pressure distribution compared to the MAU type one as shown in Fig. 5. 4. 2 Cavitation Performance

Cavitation experiments were performed at the

Marine Propeller Cavitation Tunnel of The Uni-

versity of Tokyo"). The working section is of a

square form whose side length is 450 mm. The

wake was simulated by a wire mesh screen. Cavi-

tation observation in both the uniform flow and

the wake, pressure fluctuation and noise measure-

ments in the wake were performed. The number

of propeller revolution was 32.5 rps except for the

cavitation inception test in the uniform flow which

was performed at 25 rps. Only for the experiments

in the wake, steel powder of 90-160 pm in diame-

ter, which corresponds to the roughness Reynolds

number of about 2000, was sparsely distributed

near the leading edge of the propellers in order to

reduce the cavitation intermittency. For this pur-

pose, also the air content of the water was control-led to keep the saturated condition at the test sec-

tion pressure").

The pressure fluctuations were measured by the

pressure transducers mounted on a flat plate which was installed above the model propeller as shown

in Fig. 14. The tip clearance was about 0.26 timesFig. 12 Propeller open characteristics

Fig. 13 Comparison of experimental propeller open efficiency based on Kr/J2


the propeller diameter. The pressure transducers

were fixed by silicone gum in order that the mea-

surement might not be disturbed by the flat plate

vibration. The cavitation noise was measured by

a B K 8103 type hydrophone inside the cavita-

tion tunnel. That was placed at the position of

Ψ=75.1°and r=1.535R in the propeller plane.

Tip vortex cavitation bucket charts are shown

in Fig. 15. No significant difference is observed

since all the propellers were designed to have the

same thrust distribution in the radial direction.

Figure 16 shows the sheet cavitation bucket charts.

It can be said that the experimental results appro-

ximately follow the relation predicted from the

calculation shown in Fig. 8, although the back

cavitation curve of MP 017 has higher inclination

compared with those of the others and the face

cavitation curve of MP 016 agrees with that of

MP 017.

Figure 17 shows the comparison of cavity extent

observed in the uniform flow. It is seen that the

higher design lift coefficient gives smaller cavity

as expected, MP 017 generating the cavity with

almost the same extent as that of MP 002.

Figure 18 shows the cavitation pattern in the

wake. Neither cloud nor bubble cavitation was

observed on any propellers. Regarding the relation

of cavity extent among the propellers, the same

fact as in the uniform flow can be seen.

The amplitude and phase angle distributions of

the fluctuating pressure at the 1 st and 2 nd blade

frequencies are respectively shown in Figs. 19 and

20 together with the results in the noncavitating

condition, while Figure 21 shows the comparison

of the maximum fluctuating pressure amplitude

among all the propellers and the noncavitating con-

dition. It can be seen from Figs. 19 and 20 that

the distribution shapes of the pressure amplitude

due to the cavitating propellers are almost similar

Fig. 14 Measuring apparatus for pressure

fluctuations

Fig. 15 Tip vortex cavitation bucket chart ;

11 =25rps

Fig. 16 Sheet cavitation bucket chart ;

n= 25 rps

Fig. 17 Comparison of cavity extent in uni-

form flow ; KT=O. 25, an=2. 00, n=

32. 5 rps


Direction of Rotation

1) MP002 (MAU)

2) MPO 1 6

3) MPO1 7

4) MPO1 8

one another with no significant difference in the

phase. The following results can be drawn from Fig. 21 :

( 1 ) The propeller with the lowest design lift coefficient, MP 016 does not show lower pressure fluctuation amplitude than the original MAU type

propeller MP 002. ( 2 ) The pressure fluctuations of the propeller

with the middle design lift coefficient, MP 017 is about O. 8 times as high as those of MP 002 at both the 1 st and the 2 nd blade frequencies. Regarding the 1 st one, the MP 017 value agrees approximately with the mean of the MAU type and the non-cavitating values.

( 3 ) The propeller with the highest design lift coefficient, MP 018 shows almost the same pressure

fluctuation value as MP 017 and MP 002 at the 1 st and the 2 nd blade frequencies respectively, al-though the least cavitation was generated on this

propeller. ( 4 ) Summarizing the above results, it can be

said that the adoption of the highest design lift coefficient, which is more critical to the face cavi-tation generation, is not necessary for reducing the

pressure fluctuations but the lower one is more effective especially for the higher blade frequency component. The reasons for these results will be discussed in the next section.

Figure 22 denotes the comparison of high fre-

quency noise in cavitating condition. It is seen that the increase in the design lift coefficient les-

Fig. 18 Comparison of cavity pattern in the wake ; Design Point (KT/J2=0. 552,σn/J2=6.49), n=32.5rps


1) Transverse Distribution 2) Longitudinal Distribution

1) Transverse Distribution 2) Longitudinal Distribution

sens the cavitation noise. MP 018 generates about

10 dB lower noise in the range of 10 to 100 kHz

than MP 002, while the noise due to MP 017 agrees

approximately with that due to MP 002. This re-

markable noise reduction achieved by MP 018 is

considered due to the reduction of cavity volume

which is the amount of the cavity collapsing in each

propeller revolution.

Fig. 19 Comparison of fluctuating pressure amplitude and phase angle at the 1 st

blade frequency

Fig. 20 Comparison of fluctuating pressure amplitude and phase angle at the 2 nd

blade frequency


5. Discussion on the Results of Pressure

Fluctuation Measurement

The experimental results described in the preced-

ing section showed that the pressure fluctuations

due to cavitation did not decrease consistently with

the cavity volume. The reasons for this result are

discussed in this section.

At first, let us consider the reason why MP 018

gave high pressure fluctuation at the higher blade

frequency. Figure 23 denotes the standard devia-

tion and variation coefficient of the cavity area on

each propeller obtained from 20 photographs taken

at the blade angle position of 0•‹. As exampled by

this figure, the cavity on MP 018 was far more

unstable especially in growing process compared

with the other propellers in spite of the adoption

of the leading edge roughness and air content con-

trol. This probably caused the increase in the pres-

sure fluctuations at the higher blade frequency.

This unstableness of the cavity size is considered

due to the pressure distribution characteristics.

It is inferred from this result that the flat pressure

distribution is not so effective for the suppression

of the cloud cavitation which is often generated

when sheet cavitation collapses towards the trail-

ing edge.

Secondly, let us consider the reason for the low

pressure fluctuations due to MP 017. Figure 24

shows the comparison of cavity volume obtained

by integrating the cavity thickness which was mea-

sured by a pin gauge method"). Judging from the

accuracy (•}0. 5 mm) and the number of the cavity

Fig. 21 Effect of design lift coefficient on the

maximum fluctuating pressure ampli-

tude

Fig. 22 Comparison of cavitation noise

Fig. 23 Unstableness of cavity at Yi=0° Stan-

dard Deviation and Variation Coeffici-

ent of Cavity Area on the Blade

Fig. 24 Comparison of cavity volume variation


thickness measurements, the error of about 20% is probable on the calculated cavity volume. It can be seen, however, that the cavity volume de-creases with increasing design lift coefficient. It is also seen that the cavity volume variation par-ticularly on MP 017 is smaller than that of MP 002. Since 2 nd time derivative of the cavity volume mainly affects the pressure fluctuations, it is con-sidered that the smaller time variation of cavity volume on MP 017 caused the lower pressure fluc-tuations. The reason for the reduction of cavity volume variation due to the new propellers is considered as follows : As can be inferred from the fact that the MAU type propeller MP 002 has a gently-sloping peak near the midchord in the back surface pressure dis-tribution shown in Fig. 5, the blade section of

MP 002 swells near the midchord more than those of the new propellers. Such section shape gives thinner cavity near the midchord but thicker one in the rear part, resulting in larger cavity volume

variation.

6. Conclusions

A new propeller design method to obtain the blade section shapes by prescribing pressure distri-bution was developed, combining a 2-dimensional foil design theory with a propeller lifting surface one. In this method, new blade sections can be designed in accordance with the variation of the section lift coefficient in a given wake.

Three propellers with flat pressure distribution were newly designed, changing the design lift coef-

ficient. In designing all the new propellers, only the blade section shapes and pitch were altered in order to extract the effects of the blade section improvement, keeping the blade contour, the load distribution in radial direction, etc. same as those of the original MAU type one.

Using these new and MAU type propellers, open characteristics test and cavitation experiments were

performed to clarify the effects of the design lift coefficient. The following results were obtained.

( 1 ) All the new propellers showed higher open efficiency than the MAU type one. At the design

point, the increase in the design lift coefficient raised open efficiency, the propeller with the high-est design lift coefficient showing 4. 5% higher effi-ciency compared with the MAU type one.

( 2 ) In the cavitation experiments only the sheet cavitation was observed. The cavity extent and volume decreased with increasing design lift coefficient.

( 3) Regarding the first blade frequency com-ponent, both the propellers with the middle and the highest design lift coefficients showed about

0.8 times fluctuating pressure amplitude due to the MAU type one. This value agreed approximately with the mean of the MAU type and noncavitat-ing values. On the other hand, the second blade frequency component of the propeller with the middle design lift coefficient was about 0.8 times the MAU type value, while the propeller with the highest design lift coefficient showed almost the same value as the MAU type one. This re-sult shows that the adoption of the highest design lift coefficient, which is more critical to the face cavitation generation, is not necessary for reduc-ing the pressure fluctuations but the lower design lift coefficient is more effective especially for the higher blade frequency. This result is considered due to the reduction of cavity volume variation.

( 4 ) Higher design lift coefficient gave lower cavitation noise. The cavitation noise due to the

propeller with the highest design lift coefficient was about 10 dB lower in the frequency range of 10 to 100 kHz than that of the MAU type one. This result is considered due to the reduction of cavity volume which is amount of the cavity col-lapsing in each propeller revolution.

Acknowledgments

The authors express their sincere gratitude to the staffs in Akishima Laboratories (Mitsui Zosen) Inc. and Highspeed Dynamics Laboratory, Depart-ment of Naval Architecture, The University of Tokyo for their help in conducting the experi-ments. The authors are also indebted to Mr. A. Sugatani for his assistance in developing the pro-

peller design method. The authors' gratitude is extended to Miss J. Fujimori and Mrs. M. Hirokawa for their typewriting the manuscript. The HITAC M-682 H System at the Computer Centre, The Uni-versity of Tokyo was used for the calculations in this research.

References

1) For example, Yamasaki, S. et al. : "Research on Highly Skewed Propellers (1 st and 2 nd

Report)", Naval Architecture and Ocean Engineering, SNAJ, Vol. 20 (1982), Vol. 22

(1984). 2) For example, Yazaki, A. et al.. "Open Water

Test Series with Modified AU-Type Four Bladed Models", Journal of Zosen Kyokai,

SNAJ, Vol. 108 (1960).

3) Abbott, I. H. and Von Doenhoff, A. E. : "Theory of Wing Sections" , Dover Publica-tions Inc., New York, 1959.

4) Van Lammeren, W. P. A. et al.: "The Wagen-ingen B-Screw Series", Trans. SNAME, 1969.

5) Lindgren, H. et al. : "The SSPA Standard


Propeller Family Open Water Characteris-tics", Publication of SSPA, No. 60 (1967).

6) Kadoi, H. et al. : "On the Development of the SRI.B Propellers and Those Characteris-tics", Report of Ship Research Institute, Vol. 21, No. 6 (1984) (in Japanese).

7) Takahashi, M. et al.: "The Cavitation Characteristics of MAU Type Propeller (1 st, 2 nd and 3 rd Report)", Journal of SNAJ, Vol. 141 (1977), Vol. 143 (1978) (in Japa- nese).

8) Izumida, Y.: "Study on Propeller Design Method in Cavitating Flow (1 st and 2 nd Report)", Journal of SNAJ, Vol. 155 (1984), Vol. 160 (1986) (in Japanese).

9) Yamaguchi, H. et al. : "Development of Ma-rine Propellers with Better Cavitation Per-formance (1 st Report)", Journal of SNAJ, Vol. 158 (1985) (in Japanese).

10) Yamaguchi, H. et al. : "Development of New Marine Propellers with Improved Cavitation Performance", Proc. Int. Symp. on Propeller and Cavitation, Wuxi, China, CSNAME, 1986.

11) Eppler, R. and Somers, D. M.: "A Computer Program for the Design and Analysis of Low-Speed Airfoils", NASA TM 80210, 1980.

12) Ito, T. et al. : "Calculation on Unsteady Pro-

peller Forces by Lifting Surface Theory",

Symp. on Hydrodynamics of Ship and Of-fshore Propulsion Systems, Oslo, 1977.

13) Nakazaki, M. et al. : "A Study on the Three Bladed Propellers with Smaller Blade Area Ratio Designed by the New Method", Jour-nal of the Kansai Society of Naval Archi-tects, Japan, No. 201 (1986) (in Japanese).

14) Sato, K. : "A Method to Design Propellers with Prescribed Pressure Distribution on the Blade Surface", Journal of SNAJ, Vol. 161 (1987)

(in Japanese). 15) Sugatani, A. and Inaba, K.: "Effect of Blade

Surface Pressure Distribution on Propeller Cavitation Performance", Graduation Thesis, Department of Naval Architecture, The Uni- versity of Tokyo, 1986 (in Japanese).

16) Kurobe, Y. et al. : "Measurement of Cavity Volume and Pressure Fluctuations on a Mo-del of the Training Ship "SEIUN-MARU" with Reference to Full Scale Measurement", Report of Ship Research Institute, Vol. 20,

No. 6 (1983) (in Japanese) 17) Kato, H. et al. : "New Marine Propeller Ca-

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Table A-1 Eppler's pressure distribution parameters of the designed

2-dimensional foils ; MP016


Table A-4 Blade section shapes and pitch distribution of MP 016


Table A-5 Blade section shapes and pitch distribution of MP017


Table A-6 Blade section shapes and pitch distribution of MP018


tions", Journal of SNAJ, Vol. 148 (1980).

Appendix

For those who hope to follow and improve the

present design, the authors show here the Eppler's design parameters in Tables A-1 through A-3, and

blade section shapes and pitch distributions of the

new propellers in Tables A-4 through A-6. With

regard to the meaning of the Eppler's design para-

meters, please refer to the literature"). The pres-

cribed parameters in designing 2-dimensional foils

were ƒË, ƒ¿*, ƒÊ, ƒÉ, ƒÉ*. The other parameters were

determined by the iteration calculation in the com-

puter program. The value of KH (upper)+KH

(lower) was used to control the foil thickness

ratio.

development of marine propellers with better cavitation

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