INFLUENCE OF WAKE ON PROPELLER LOADING
A N D CAVITATION
by
163
INFLUENCE OF WAKE ON PROPELLER LOADING
AND CAVITATION
by
W . VAN G E N T - ) and P. VAN OOSSANEN')
Abstract
In this paper an attempt is made to determine some aspects of the interrelationship between Wakefield, adapted propeller design, cavitation and dynamic load. This is p e r f o r m e d by meansof theoretical procedures f o r propeller design, f o r the determination of the dynamic propeller load and f o r the determination of cavitation performance. The use and need f o r such theoretically based calculation methods,in addition to experimental methods is discussed.
A n u m b e r o f f i v e - h o l e pitottube measurements, p e r f o r m e d atthe N . S. M . B . , are analysed to deter-mine whether or not i t is possible to derive properties of wakefields which are typical f o r certain types of ships. This is done f o r a group of large tankers and some container ships. F o r these two ships two p r o p e l l e r series are designed. F o r the tanker a series of which the blade number is v a r i e d and f o r the container vessel a series having a v a r y i n g margin against cavitation. The m e r i t s of the conventional design procedure, typical of procedures in present use, are c r i t i c a l l y reviewed. The combinations of nominal wakefields and designed propellers are analysed with respect to the dynamic load due to the inhomogenity of the vvakefield and the occurrence and extent of cavitation. Use i s made of calculation programmes based on unsteady l i f t i n g surface theory f o r the dynamic load and based on quasi-steady propeller theory combined with two-dimensional wing theory f o r the cavitation prediction.
The main conclusions are that i n order to calculate accurately the dynamic load distributions very detailed knowledge of the wake s t r u c t u r e i s r e q u i r e d and that it is necessary to incorporate cavitation m i n i m i s a t i o n c r i t e r i a i n propellers design procedures.
1. Introduction
In the last decennia, a large growth can be detected i n a l l kinds of technologies. In the f i e l d of ship design and shipbuilding this development has led to research, on the model scale, in more fundamental hydrodynamics and mechan-i c s . Now, not only the results of r e l a t mechan-i v e smechan-imple resistance and propulsion tests are r e q u i r e d , but also such basic i n f o r m a t i o n as mechanical stresses and vibrations and the s t r u c t u r e of hydrodynamical velocity f i e l d s i n t h e neighbour-hood of the h u l l . To f a c i l i t a t e such r e s e a r c h , both theoretical knowledge and advanced e x -p e r i m e n t a l equi-pment are r e q u i r e d . In the f i e l d of ship propulsion this has led to the building of depressurized towing tanks and large c i r c u l a t ing water channels p r i m a r y f o r the study of i n -t e r a c -t i o n phenomena be-tween ship hull and p r o p e l l e r , inclusive of the influence of c a v i t a -tion.
•) Research Scientists, Propulsion Hydrodynamics Department, Nether-lands Ship Model Basin, Wageningen, The NetherNether-lands.
A p r i n c i p a l d i f f i c u l t y s t i l l involved i n the i n t e r p r e t a t i o n of model experiments is the e x i s t -ence of scale effects. The importance of scale effects increases with the increasing size of ships. An important example is the effect of Reynolds number. To overcome the d i f f i c u l t i e s associated with scale effects i n p r e d i c t i n g f u l l scale behaviour, basic theoretical knowledge of the phenomena concerned is necessary. In a d -d i t i o n , the -development of a-dequate theory is of •importance in the development of balanced
design procedures. It should be noted that the use of theoretical methods i n this manner does not compete w i t h experimental methods. The basic i n f o r m a t i o n f o r the development of theoretical procedures can only be obtained e x -p e r i m e n t a l l y .
In this paper the use of three t h e o r e t i c a l procedures is demonstrated. An attempt is made to show the i n t e r r e l a t i o n s h i p between the s t r u c t u r e o f t h e vvakefield i n which the p r o p e l l e r
1 6 4
works and the r e s u l t i n g unsteady load charac-t e r i s charac-t i c s and cavicharac-tacharac-tion performance. Use is made of a n u m e r i c a l programme f o r propeller design, f o r dynamic p r o p e l l e r load calculation and f o r cavitation p r e d i c t i o n .
2. Analysis of wakefields 2.1. General remarks
In the analysis of the hydrodynamical action of the screw propeller behind a ship an i n c r e a s ing amount of attention is being given to the e f fects of the nonhomogeneous inflow. The i n -homogenity is the result of the presence of the ship's h u l l , which deforms the o r i g i n a l s t r e a m -lines and causes a retardation of the relative stream velocities due to viscous action. In a d -dition, the v i c i n i t y of a f r e e surface w i t h gravity waves and the occurrence of flow separation can not only influence the inhomogenity but also i n -troduce an unsteadiness of the wakefield behind the ship.
As long as the action of the propeller is not incorporated.in the picture of the wal<efield i t is called the nominal wakefield. This nominal wakefield i s used as a s t a r t i n g point i n p r o p e l -l e r ana-lysis. I t is obvious that the s t r u c t u r e of the wakefield at the location of the p r o p e l l e r is described i n t e r m s of the f r a m e of reference
most suitable to the p r o p e l l e r ; that i s a system of c y l i n d r i c a l coordinates. The variations of the wakefield i n the axial d i r e c t i o n are assumed to be s m a l l and are usually neglected i n the analysis.
More important i n the analysis are the v a r i a -tions of w a k e f i e l d properties in r a d i a l d i r e c t i o n . It has to be noted that these properties also vary w i t h the angular position. F o r a long t i m e only the averages of the p e r i p h e r a l l y v a r y i n g quantities w e r e considered whereby the r a d i a l distributions are distributions of p e r i p h e r a l averages, A consequence of this r e s t r i c t i o n is that only steady phenomena appear i n the analysis.
Unsteady phenomena appear as soon as the non-homogeneous structure of the wal<:efield is considered m o r e completely. The peripheral variations are the cause of unsteady cavitation and dynamic loads on blades and, when v e c t o r -i a l l y composed, on the complete p r o p e l l e r . In addition to these phenomena on the p r o p e l l e r i t s e l f , the v e l o c i t y and pressure f i e l d
surround-ing the propeller and the hydrodynamical action on neighbouring bodies such as rudders, the ship's h u l l , other p r o p e l l e r s , etc. . can only be analysed i f t h e complete picture of the wakefield is considered. In these l a t t e r cases an essential f a c t o r i n the unsteady behaviour is the effect of the finite number of p r o p e l l e r blades c o n t r a r y to the effects on the blades themselves which do not depend on the number of blades.
22. Experimental methods
F o r a l o n g t i m e the only property o f t h e w a k e f i e l d which was considered, was the axial c o m -ponent of the velocity f i e l d . This was p a r t l y due to the fact that the existing analysing theories and computer programmes were not adapted to incorporate tangential and r a d i a l velocity c o m -ponents. Besides this reason, presumably more causal was the f a c t that w i t h the available e x perimental tools ( f o r instance the Prandtl p i t o t -tube) only one component of a velocity vector in a pre-assumed d i r e c t i o n could be measured.
At this t i m e we are i n the position to p e r f o r m measurements by means of a f i v e - h o l e pitottube, w i t h which i t is possible to estimate the three components of an a r b i t r a r i l y d i r e c t e d v e l o c i t y vector between certain l i m i t s . This p o s s i b i l i t y has stimulated the f u r t h e r development of the analysing theories and computation procedures to cope w i t h the additionally obtained i n f o r m a -t i o n of -the veloci-ty f i e l d .
I t can be expected, f u r the r m o r e , that i n the near f u t u r e the analysis w i l l proceed to the stage that experimental knowledge is r e q u i r e d concerning the influence of the w o r k i n g p r o p e l l e r on the nominal w a k e f i e l d (directly and i n d i r e c t l y via p r o p e l l e r h u l l interaction). In addition the u n steady aspects of the wakefield due to t u r -bulence and unsteady flow separation becomes of interest. At this moment there are two e x -perimental techniques in development w h i c h can possibly he adapted to these specific demands.
One method is the application of the h o t f i l m anemometer. The sensor of this system can be mounted i n a manner s i m i l a r to the o r d i n a r y and the f i v e - h o l e pitottube. In this way velocity measurements can be p e r f o r m e d i n a number of positions and the obtained signals decomposed i n steady and unsteady parts, giving a t o t a l p i c -ture of the nominal w a k e f i e l d . More i n t e r e s t i n g is the configuration in which the sensor is
165
mounted before the w o r k i n g propeller and either n o t r o t a t i n g or rotating with the p r o p e l -l e r . In the f i r s t case the actua-l w a k e f i e -l d is measured inclusive of the unsteady action of the propeller due to i t s f i n i t e number of blades. In the case i n which the sensor rotates w i t h the propeller the velocities are measured as they are experienced by the blades, whereby the i n homogenity of the wakefield is d i r e c t l y t r a n s l a t -ed into its time-dependent action on the blades. A disadvantage of the h o t f i l m sensors is that they need frequent c a l i b r a t i o n .
Another p r o m i s i n g experimental method is the l a s e r - D o p p l e r anemometer. This method has the advantage that i t does not d i s t u r b the f l o w at the location of the measurements. The requirement, however, that the laser beam must be able to reach each point of measure-ment and that i t s dispersion must be received again, may be d i f f i c u l t to r e a l i z e i n a l l cases.
It should be r e a l i z e d , f u r t h e r m o r e , that these experimental methods are used to determine the wakefields of ship models of which the c o r r e l a -tion w i t h the f u l l scale has as yet to be proved. In this connection the idea of the screw i t s e l f being a measuring instrument may be useful f o r f u l l scale measurements. The blades of the p r o p e l l e r have to be provided with a number of pressure transducers and s t r a i n gauges. By means of hydrodynamical and mechanical theory the r e l a t i o n with the actual w a k e f i e l d can be provided and the w a k e f i e l d determined, and the result compared w i t h model measurements.
2.3. Analysing procedures
An investigation into the c o r r e l a t i o n between the s t r u c t u r e of the w a k e f i e l d and the hull shape can be divided into two stages. In the f i r s t stage a global approach can be used, which means that c o r r e l a t i o n is sought between the w a k e f i e l d structure and the type of ship indicated by d i s placement, b l o c k c o e f f i c i e n t and type of a f t e r -body. Depending on the r e s u l t s of this stage i t must be judged whether or not a r e f i n e m e n t of the c o r r e l a t i o n has sense and is possible i n a second stage. In this second stage a more de-tailed description of the hull shape i s r e q u i r e d in t e r m s of p a r a m e t e r s which have a d i r e c t r e l a t i o n w i t h the s t r u c t u r e . Besides the choice of h u l l p a r a m e t e r s , the choice of w a k e f i e l d parameters i s also i m p o r t a n t . F o r p r a c t i c a l
purposes one can r e s t r i c t the interest to the nominal w a k e f i e l d at the location of the p r o p e l -l e r . The usua-l presentation is by means of a p e r i p h e r a l d i s t r i b u t i o n of each of the three velocity components, a vector d i a g r a m of the transverse velocity and a plot of lines of equal longitudinal velocity. A d i f f e r e n t method is to decompose the p e r i p h e r a l distributions by means of harmonic analysis and to present the r a d i a l d i s t r i b u t i o n s of harmonic amplitudes and phase angles.
In 1965 Hadler and Cheng [1] *) delivered an analysis of experimental wake data in way of the p r o p e l l e r plane of single and t w i n - s c r e w ship models. One of t h e i r aims was to attack the above-mentioned problems of c o r r e l a t i o n between wake and h u l l . In the present paper a s i m i l a r analysis is applied to a group of large tankers with block coefficients beyond the range of Ref. 1 . Table 1 gives the p a r t i c u l a r s of this group of tankers. The a i m was to investigate, whether o r not general r e m a r k s could be made about the wakefields o f s u c h a group and whether or not an average f i e l d could be defined. In other w o r d s : how large is the scatter i n the e x -perimental i n f o r m a t i o n obtained by means of f i v e - h o l e pitottube measurements. A f u r t h e r a i m of this paper, depending on the results of the f i r s t a i m , was to exercise w i t h such an average f i e l d to determine some general r e m a r k s about the choice of the p r o p e l l e r f o r the u n d e r l y i n g group of ships.
2.4. Examples of wakefield types
As stated i n the foregoing section one of the aims of t h i s paper was to f i n d out whether o r not i n the available threedimensional w a k e -f i e l d s o-f some groups o-f ships some common f a c t o r s could be recognized. Special attention was given to the harmonic analysis of the p e r i p h e r a l velocity d i s t r i b u t i o n s at various r a d i i . It is often a point of discussion whether or not the accuracy of the higher harmonics (of order equal to the blade number) i s s u f f i c i e n t . An answer to this question is p a r t l y given when the harmonic content shows significant i n f o r m a -tion i n the amplitudes of the higher harmonic components.
In the f o l l o w i n g the harmonic analysis of the •) Numbers in bracltets refer lo tlie list of references at ttie end of this paper.
1 6 6
Table 1.
Value of ship parameters f o r which tanker wakefields are presented
Length between perpendiculars L p p 300 to 375 m
Moulded beam B 50 to 62 m
Draught T 20 to 27 m
Displacement A 300000 to 500000 m"^
Midship section coefficient jJ 0. 995 to 0.9975
P r i s m a t i c c o e f f i c i e n t 9 0, 82 to 0.845
Longitudinal centre of buoyancy
3.4% L p p
behind midship section L C B 2. 5% L p p t o 3.4% L p p
Ship speed i n loaded condition V 15 to 17 knots
Power at P r o p e l l e r ( m e t r i c ) P 30000 to 50000 HP
R P M of p r o p e l l e r N 80 to 95
Diameter of p r o p e l l e r D 8. 5 to 9.2 m
three velocity components is represented i n the f o l l o w i n g way:
10
V ^ / V and V^JV = A q " ^ f A ^ c o s ( m q ) - p ^ ) 10
V ^ / V = 2 A ^ s i n ( m 9 - P j ^ )
\ f ' ^ y ' ^ were available. Of a l l these wakefields an h a r
-X monic analysis of the three velocity components
where <f is the angular position i n clockwise d i r e c t i o n (bottom position is z e r o ) .
2.4.L Single screw tankers (conventional stem)
The f i r s t group of ships to which attention was given was the group of l a r g e , single screw t a n -k e r s w i t h conventional sterns. F r o m the f i l e s o f t h e N . S . M . B . eleven models were chosen of
i!^ which complete t h r e e - d i m e n s i o n a l wakefields
was p e r f o r m e d and the r e s u l t i n g amplitudes were plotted as a f u n c t i o n of the radius r a t i o . Due to the assumption that the w a k e f i e l d of a single screw ship is s y m m e t r i c a l , the phase angles of the harmonic components can be r e -presented by giving a sign to the amplitudes.
A f t e r a close inspection of these r a d i a l d i s -t r i b u -t i o n s i -t became obvious -tha-t f r o m -the group of eleven models a subgroup could be s p l i t t e d o f f ; namely those w i t h a closed screw aperture (with sole). Table 1 gives the p a r t i c u l a r s of these models. T h i s subgroup of f i v e models showed a better mutual agreement than the r e m a i n i n g six models w i t h open screw a p e r t u r e . W i t h i n due t i m e i t was not possible to give an explanation f o r the l e s s e r agreement of t h i s l a s t group o r to c o r r e l a t e the d i f f e r e n c e s to obvious g e o m e t r i c a l v a r i a t i o n s such as the 'cutaway' of the s t e r n below the p r o p e l l e r shaft.
i.or E < i 0 . 5 h m " 0 T A N G E N T I A L I S D E F I N E D T O B E Z E R O F O R 0 . 1 0 S Y M M E T R I C A L S H I P S W A K E F I E L D 1 • 2 1 3 n 4 a 6 * 0 . 0 5 R A D I A L _ L m • O o a o 1_ 0 . 5 1.0 0 5
Figure 1. Radial distribution of average velocity components of tanker wakefields (harmonic m - 0) 1.0
173
Figure 8. Linesüfequallongitudinal velocity tor average tanker wakefield.
The results of the harmonic analysis of the f i v e tanker wakefields w i t h closed screw a p e r -t u r e are presen-ted i n Figures 1 -to 6. A x i a l , tangential and r a d i a l components of the same harmonic order are given side by side. Although a numberof graphs show a r e l a t i v e large scatter of points, there are others which show a very good c o r r e l a t i o n . F o r every harmonic c o m -ponent an avex'age l i n e is drawn as a function of the r a d i u s . The shape of these lines is not a
mere average of the individual points, but more an average of the lines f o r e a c h model. In f u t u r e work a w e l l - d e f i n e d rule f o r such an averaging process is needed.
Especially the lines f o r the lower harmonic components have a shape, which correspond to an obvious physical meaning. F o r example, the zero axial harmonic (average axial velocity) can be integrated to f i n d the v o l u m e t r i c flow rate w i t h i n each radius. The zero tangential h a r monic is identically zero due to the m i r r o r s y m m e t r y of port and starboard parts of the w a k e -f i e l d . The shape o-f the -f i r s t tangential harmonic can be explained as corresponding to the e x i s -tence of two v o r t i c e s on both sides of the plane of s y m m e t r y .
An i m p r e s s i o n of the usefulness of the average r a d i a l d i s t r i b u t i o n s of the harmonic components can be obtained by reproducing the three components of the v e l o c i t i e s . These are achieved when the summation over the eleven harmonic components is p e r f o r m e d . The r e s u l t s are given i n F i g u r e 7 where the p e r i p h e r a l d i s -t r i b u -t i o n s of -the -three veloci-ty componen-ts are given f o r various r a d i i . The picture is r e a s o n -able. F i g u r e 8 gives the plot of lines of equal longitudinal velocity.
Now the question a r i s e s : what is the value of such an average f i e l d ? Is i t useful as a t y p i c a l wakefield f o r this group of ships? The answer to these questions can only be obtained by i n -vestigating the effects of the deviations around the average on the various p r o p e l l e r p r o p e r t i e s .
2.4.2. Twin-screw container ships
The second group of ships to which attention was given were t w i n - s c r e w t h i r d generation
container vessels. F r o m the f i l e s of the A X I A L E < iO.St-m - O 0 . 5 0 0 , 0 5 m - O T A N G E N T I A L W A K E F I E L D 6 o 7 • 1,00 aso 1,00 0 , 2 , 0,1 m = 0 -1-0 . 5 -1-0 r / R ' r / R r / R
Figure 9. Radial distribution of average velocity components of container vessel wakefields.
Table 2.
Value of ship parameters f o r which container ship wakefields are presented
Length between perpendiculars L p p 220 to 260 m
Moulded beam B 30 to 32 m
Draught T 9 to 10 m
Displacement A 35000 to 40000 m"^
Midship section coefficient p 0. 92 to 0. 93
P r i s m a t i c coefficient 9 0. 55 to 0. 59
Longitudinal centre of buoyancy
i n f r o n t of midship section L C B 1.37p L p p t o 2.4% L p p
Ship speed i n loaded condition V 28 to 32 knots
Total power at p r o p e l l e r s ( m e t r i c ) P 60000 to 120000 HP
RPM of p r o p e l l e r s N 130 to 175
Diameter of p r o p e l l e r s 6.2 to 7. 5 m
N . S. M . B . six models were chosen of which the complete threedimensional wakefield m e a s u r e -ments were available. Geometrical parameters which have a strong influence on the wakefield s t r u c t u r e at the location of the p r o p e l l e r s are the distance between the propeUer shafts (es-pecially when the distance is s m a l l ) , the presence of shaft supporting s t r u t s (of which the shape and position strongly vary f r o m model to model) and the shaft i n c l i n a t i o n . Consequent-l y , the resuConsequent-lts of the harmonic anaConsequent-lysis showed a large scatter. The graphical representation of these results is more complicated than f o r s i n -gle screw ships due to the f a c t that any value of the phase angles can occur.
For two of these models the results of the harmonic analysis are presented i n Figures 9 to 14, to give an idea of the differences which can occur. Table 2 gives some p a r t i c u l a r s of these models. F i g u r e 15 gives the p e r i p h e r a l velocity d i s t r i b u t i o n s of one case (wakefield no. 6). In chapter 4 the effects of the differences w i l l be studied.
3. Particulars of propellers and propeller design procedure
3 J . Basic considerations
As stated i n the introduction, the purpose of this study was to determine how the ship's wake influences the unsteady f o r c e and moment properties and the cavitation performance of the screw p r o p e l l e r . I n t h i s context i t is also pos-sible to determine whether o r not other values of one or more p r o p e l l e r parameters can
pos-sibly i m p r o v e these p r o p e r t i e s . Standard p r o p e l l e r design practices, at this t i m e , s t i l l incorporate c r i t e r i a which have been developed f r o m considerations i n u n i f o r m flow o r , a t t h e most, i n a r a d i a l l y v a r y i n g i n f l o w . No detailed p r o p e l l e r design method is as yet available which ensures a c o r r e c t i n t e g r a t i o n of p r o p e l l e r geometry w i t h the p e r i p h e r a l v a r y i n g inflow velocities such that o p t i m u m hydrodynamic p r o p e l l e r properties are obtained. It stands to reason that the choice of, e . g . the chord length of the blade sections by means of the c r i t e r i u m of avoiding cavitation at the s h o c k - f r e e angle of attack, w i l l generally not represent an o p t i m u m value f r o m the point of view of c a v i t a t i o n sup-pression at non-shock f r e e angles of attack. The design of a p r o p e l l e r f o r the t y p i c a l w a k e f i e l d f o r large tankers w i t h a conventional s t e r n (see F i g u r e 7) and f o r a given w a k e f i e l d f o r a f a s t t w i n - s c r e w 3 r d generation-type container vessel (Figure 15)was, t h e r e f o r e , p e r f o r m e d according to standard practices f o r these vessels, to a s -sess i n this way the value of the v a r i o u s c r i t e r i a incorporated i n these conventional p r o p e l l e r designs.
3.2. Description of propeller design procedure
The standard propeller design procedure available at the N . S. M . B . i s based on the L e r b s induction f a c t o r method [2]. W i t h t h i s t h e o r y the induced v e l o c i t i e s follow f r o m the w e l l - k n o w n f o r m u l a s :
181 U 1 dx A 1 , dG 0_ V~ 2 l A d x " x - x A Ü O T V dG dx X, T dx ( 1 ) (2) in which U ^ , U ^ = a x i a l , respectively, tangential i n -duced velocity due to the t r a i l i n g vortex sheets at the l i f t i n g l i n e , V . = average, nominal axial wake veloc¬
i t y ,
Xj^ = dimensionless hub radius,
I ^ , Ip = a x i a l , respectively, tangential i n -duction f a c t o r and
^ d x = dimensionless c i r c u l a t i o n of t r a i l -dx„ o
" ing v o r t e x at x^.
The average, nominal a x i a l wake velocity is determined by means of the r e l a t i o n :
ƒ dx J- V^xdq> (3)
where
= measured a x i a l wake v e l o c i t y component at ( x , 9 ) .
The induction f a c t o r s 1 ^ and are dependent on the r a t i o X / X Q , the hydrodynamic pitch angle P j and the number of blades z. Wrench [ 3 ] has derived a set of f o r m u l a s f o r the calculation of these induction f a c t o r s .
Between the induced velocities and U-pand the hydrodynamic pitch angles p and P j , defined as shown i n F i g u r e 1 6 , the f o l l o w i n g equation can be d e r i v e d :
in which
= l i f t coefficient at x and,
c = chord length of blade section at x , V f j = resultant velocity at x ,
D = propeller diameter and n = rotative p r o p e l l e r speed.
On combining equations 5 and 6 the f o l l o w i n g r e l a t i o n is obtained:
C • c 2TTG cosp
L* 1
D (7)
tan p V a
The available computer programme at the N . S. M . B . starts w i t h either a set of values f o r P i or f o r G i n the points x = 0 . 2 ( 0 . 1 ) 1 . 0 . F o r the design cases at hand i t was decided to use Pj-values according to the Van Manen c r i t e r i u m f o r o p t i m u m e f f i c i e n c y of p r o p e l l e r s i n a r a d i a l -ly v a r y i n g wake [ 4 ] . v i z . : 1 tanp = — 1-1 - « ( X ; where ^ TTxnD in which V = ship speed 5 = the T a y l o r wake f r a c t i o n ,
K>(x) - the T a y l o r wake f r a c t i o n at radius x defined as: 3 / 4
tanp
(8) " ( X ) = where V - V ^ ( x ) V ( 1 0 ) ^ ^ t a n p 1 = -, 0 t a n p j tanp ( 4 )The r e l a t i o n between l i f t c o e f f i c i e n t and the dimensionless c i r c u l a t i o n G i s : 2TTD V ^ where V . R XTTnD-U„ cos Pj ( 5 ) (6) ( 1 1 )
Figure 16. Definition of hydrodynamic angles and velocities of a propeller blade section.
1 8 2
Va is the measured a x i a l wake velocity c o m -ponent at (x, 9 ) .
In the i n i t i a l design stage, only a f i r s t ap-p r o x i m a t i o n to the ideal ap-proap-peller e f f i c i e n c y n j is known. I t i s , t h e r e f o r e , necessary to p e r f o r m the hydrodynamic p a r t of the p r o p e l l e r design f o r at least 3 values of ni and to calculate the
ideal thrust loading coefficient Crj,^^ and ideal
power loading coefficient C p j f o r each value of r i j . It then becomes possible by comparing the required C p h j or C p j value with the c o r r e s p o n -ding values calculated f o r the n j - v a l u e s to determine (by interpolation) the Tjvalue c o r -responding to either the r e q u i r e d C-pj^j or Cp^ value. In the standard N . S . M . B . p r o p e l l e r design programme, the three nj-values f o r which this procedure is p e r f o r m e d are the value f o l l o w i n g f r o m the K r a m e r d i a g r a m [5] corresponding to the r e q u i r e d ideal thrust l o a d -ing coefficient C p h j or the ideal power load-ing coefficient Cpj.and the values obtained by d i v i d -ing this K r a m e r value by 0.95 and 1.05. The ideal thrust loading coefficient and the ideal power loading coefficient are calculated by the relations: 1 CThi 4z ƒ G ( l - c o ( x ) ) ^ ( - ^ - = ^ ) dx (12) X h tanp v ^ and C P i 1 3 U , G O ^ ) A 4 '""P ^ (13) It f o l l o w s f r o m these equations t h a t t h e h y d r o -dynamic part of the propeller design process (which consists of determining the values of the non-dimensional c i r c u l a t i o n G and the induced velocities and U^p) must be p e r f o r m e d f o r each of the three assumed ii j - v a l u e s i n o r d e r to calculate the ideal thrust and power values. The r e q u i r e d C p ^ j or Cp^ value is determined f r o m the r e l a t i o n s : 8 T , -•Thi 2 2 •pV D (14) and B P , ^Pl 3 2 TTPV D (15) where T j ^- ideal ( f r i c t i o n l e s s ) t h r u s t , and P j = ideal ( f r i c t i o n l e s s ) power.
A f t e r specifying the values of Pj atthe various r a d i i by means of r e l a t i o n 8, the values f o r the c i r c u l a t i o n G at the respective r a d i a l stations are calculated by means of the r e l a t i o n which arises when substituting relations 1 and 2 i n 4 , v i z . : 1 J X, dG d T " I H a n p j . I dx x - x = 2 t a n p t a n p -1 (16)
The solution of this equation f o r G i s p e r -f o r m e d i n the manner given by L e r b s [ 2 ] , Each of the induction f a c t o r s and also the c i r c u l a t i o n function is expanded i n a F o u r i e r series w h e r e -by equation 16 reduces to a set of m l i n e a r equations f o r m values of the F o u r i e r c o e f f i -cients of G f o r m r a d i i values. Once these m F o u r i e r coefficients of the F o u r i e r expansion f o r t h e spanwise c i r c u l a t i o n d i s t r i b u t i o n f u n c t i o n are known, the actual values of G at the various r a d i a l stations are easily determined.
When the f i n a l values f o r Pj, G, U ^ , and C / D are determined, the procedure f o r determining the blade geometry can be s t a r t e d . Conventional design procedures at this t i m e only incorporate c r i t e r i a f o r s a t i s f a c t o r y strength and cavitation properties at the design (or shock-free) angle of incidence of the f l o w . In the N . S . M . B . design p r o g r a m m e , the e x p r e s -sion f o r the section modulus at a section XQ i s :
W ( X o ) 2 2 5 PTT n D i o r p C O S E
( ^ )
cos ( p j - P ) 2 c o s p c o s ( P i ^ - P j ) x (x-XQ)dx (17) i n which W ( X Q ) = r e q u i r e d section modulus at x ^ ,CJ m a x i m u m allowable tensile stress less the tensile s t r e s s caused by c e n -t r i f u g a l f o r c e s i n k g f / m ^ ,
£ = rake angle and Pio = Pi-value at x ^ .
183
type of thickness d i s t r i b u t i o n and camber d i s -t r i b u -t i o n applied. A-t -the N . S . M . B . , frequen-t use is made of a modified Walchner 'set B thickness d i s t r i b u t i o n ! 6] and a parabolic camber l i n e . The cavitation properties of this type of p r o f i l e are very satisfactory and very d i f f i c u l t to improve upon. The o r i g i n a l Walchner 'set B thickness d i s t r i b u t i o n was modified p r i m a r i l y f o r manufacturing purposes by means of a numerically controlled propeller m i l l i n g machine, I t w a s found necessary to adopt s l i g h t -ly t h i c k e r leading and t r a i l i n g edges f o r blade sections near the t i p where the m a x i m u m blade thickness i s s m a l l . This has also been found to be b e n i f i c i a l f r o m a cavitation point of v i e w . In p r i n c i p l e , t h e r e f o r e , a standard N . S . M . B . p r o -p e l l e r design, which incor-porates this modified Walchner s e t B thickness d i s t r i b u t i o n , w i l l have a dimensionless thickness d i s t r i b u t i o n which slightly varies w i t h p r o p e l l e r r a d i u s . F u r t h e r -more, f r o m the hub to about x = 0.5 a standard Gutsche [7] type of p r o f i l e is applied which has its p o i n t o f m a x i m u m thickness situated c l o s e r to the leading edge. This is done f o r obtaining a higher p r o p e l l e r e f f i c i e n c y . This type of p r o f i l e can attain lower drag to l i f t r a t i o s . The coordinates of the dimensionless thickness and camber d i s -tributions obtained i n this way, as a f u n c t i o n of r a d i u s , are given i n appendix A . It w i l l be n o t i c ed that the modified Walchner thickness d i s t r i butions ( f r o m about x 0. 6 to x = 1. 0) are p r a c -t i c a l l y e l l i p -t i c a l f r o m -the leading edge -to -the position of m a x i m u m blade section thickness.
F o r this modified Walchner ' s e t B ' type of thickness d i s t r i b u t i o n and a parabolic camber l i n e , the expression f o r the m i n i m u m pressure c o e f f i c i e n t at s h o c k - f r e e entry of the f l o w i s :
/ V ° \ t
/ ^ ' l " ^ - { 0 . 6 2 2 + 2 . 7 4 ) - - ( 0 . 2 n \ t Jc \ [ + 1.04 ( - ) (18) in which Cp . = m i n i m u m pressure c o e f f i c i e n t and t = m a x i m u m thickness o f t h e blade s e c -t i o n .On solving equation 18 f o r the r e q u i r e d chord length c, the f o l l o w i n g equation i s found:
O . S l l C ^ - c+1.37t 311 C l ' c+1.37t) 2 2 + Cp . (0.2(C^ • c) +1.04t ) ^ m i n L m j n in which ka (19) (20) ^ min where k = m a r g i n against cavitation; k < 1 f o r a positive m a r g i n against cavitation and k > 1 f o r a negative m a r g i n against cavitation and
a = cavitation number.
The value of the cavitation number f o l l o w s f r o m : ci(x) where V P - P -xRy O V (21) V
tanp " V a
( l - w ( x ) ) R cospi In equation 2 1 : (22)PQ = static pressure at centre line of p r o p e l -l e r shaft,
Py = vapour pressure and y = specific g r a v i t y of f l u i d ;
Equation 19 must be solved together w i t h a r e l a t i o n f o r the m a x i m u m thickness of the blade section t to obtain the r e q u i r e d strength. The relation i n use i n the N . S . M . B . design p r o -gramme i s : t ^ . c where W(x) 0 . 0 8 7 C O S (23)
W(x) ^ section modulus at x and £ = rake angle.
F r o m relations 19, 20, 2 1 , 22 and 23 the m i n -i m u m requ-ired values f o r the chord length c and blade section thickness t at the v a r i o u s p r o p e l
-l e r r a d i i are ca-lcu-lated. To obtain a feasib-le blade contour i t i s necessary to modify the r e -sulting values f o r c by a f a i r i n g procedure. This also holds f o r the r a d i a l distribution of the maximum blade section thickness t .
The geometric camber of the blade sections is determined by means of the r e l a t i o n :
K • C -c c L 4TT (24) where K = camber c o r r e c t i o n f a c t o r f o l l o w i n g f r o m c l i f t i n g surface theory.
The geometric camber at the various blade sections f o r the present designs were f u r t h e r -more corrected f o r cascade effects by means of dividing f^^ i n f o r m u l a 24 by C ^ , where:
2
= 1+0. 4TT ( t / c + ( t / c ) + — ) (25)
Application of this c o r r e c t i o n , devised by J o o s s e n [ 8 ] , is no longer standard N . S.. M., B . practice however. The value f o r K^, used i n the computer p r o g r a m m e i s analytically determined f r o m polynomials which were developed f r o m c r o s s - f a i r i n g the values given i n Ref. [9] f o r zero skew. The coefficients and t e r m s of these polynomials are given i n appendix B .
To the hydrodynamic pitch angle Pj a s m a l l c o r r e c t i o n t e r m i s added when d e t e r m i n i n g the geometric pitch. T h i s is p e r f o r m e d i n a c c o r -dance w i t h N . S . M . B . experience f o r obtaining better cavitation p r o p e r t i e s . The pitch angle
Y(X) becomes:
=
W
t
where
°^NT = 0 4 Z - 0 . 0 6 ) C l
the nominal pitch at x f o l l o w s f r o m : Pj,j =iTDtanY
The average nominal pitch i s : 1 1 N 0 . 4 8 (26) (27) (28) (29) X h
The expanded blade area r a t i o i s :
^E^^^o " 0. 5HD J c dx
X h
(30)
The delivered thrust i s :
T = j p D z J c ( C cosp - C sinp )dx (31) ^h
and the r e q u i r e d torque i s : 1 2 " ^ 2
Q = - pD z .ƒ V C ( C sinp + C cosp ) x dx
^ X h ^ L ' ° (32)
The open-water efficiency i s : T . V ( l - w )
2TTQn (33)
A f t e r the detailed geometry i s f i x e d , a moderate amount of skew is applied due to the favourable e f f e c t of skew on the instationary p r o p e l l e r load and on the cavitation p e r f o r m -ance [ 1 0 ] . A check i s necessary to ensure that the appropriate values f o r K^, were applied to determined the geometric camber. Skew not only influences the K ^ - v a l u e , but also the ideal angle of attack. On using a s y m m e t r i c a l camber l i n e , however, the ideal angle of attack i s z e r o , and the effect of skew i s l i m i t e d to the camber c o r r e c t i o n f a c t o r K„, which e f f e c t i s s m a l l f o r skews s m a l l e r than 30 degrees at the t i p .
The designed p r o p e l l e r s f o r the present study a l l have a skew of 20 degrees at the t i p . The applied r a d i a l skew d i s t r i b u t i o n i s according to the f o r m u l a :
o s = r a r c s i n [xsin20 ] where
s = distance f r o m skew line to generator i n expanded view and
r = local p r o p e l l e r radius.
The r e s u l t i n g p r o p e l l e r blade contours are shown i n F i g u r e 17 and F i g u r e 18.
3.3. Parliculars of designed propellers
F o r the t y p i c a l w a k e f i e l d of a l a r g e tanker w i t h a conventional s t e r n (see F i g u r e 7), a 4 , 5 and 6-bladed p r o p e l l e r was designed by means of the above-described procedure. The input f o r these designs are as shown i n Table 3, The strength demands were kept constant f o r a l l 3 p r o p e l l e r s . The cavitation c r i t e r i u m was that k - 0.70 (see equation 20). T h i s was also kept
185 0 7 0 2
( ( ( \
Ov z . 4 \ \ V< 6\\\ \
\\\ \
\\\ \
\\\ \
/
Figure 17. Blade contours of designed propellers for large tanker.
constant f o r the 3 tanlter p r o p e l l e r designs. Details of the r e s u l t i n g p r o p e l l e r s are given i n Table 4. The values of the drag c o e f f i c i e n t C-q were set equal to 0. 01 at a l l r a d i i .
The p r o p e l l e r designs f o r the 3 r d generation container vessel were based on wakefield 6 (see Figure 15). F o r these designs the strength demands were kept identical to those of the tanker p r o p e l l e r s . The cavitation c r i t e r i u m was v a r i e d however, k i n f o r m u l a 20 was chosen to be equal to 0.7, 0.85 and 1.0. I n this way a blade area r a t i o v a r i a t i o n was obtained. The respective ex-panded blade area r a t i o s which resulted were 1.210, 1.060 and 0.950 r e s p e c t i v e l y . The values of other input variables f o r these p r o p e l l e r designs are given i n Table 5. Details of the r e s u l t i n g p r o p e l l e r s are given i n Table 6. The values of the drag c o e f f i c i e n t s Cj^ were set equal to 0. 01 at a l l r a d i i . F i n a l l y , i t should be kept i n mind that the r a d i a l d i s t r i b u t i o n of pitch and camber of these 6 p r o p e l l e r designs, as given i n Tables 4 and 6would n o r m a l l y , at the N . S . M . B . under-go a f i n a l f a i r i n g procedure.
The 4-bladed tanker p r o p e l l e r is designated as p r o p e l l e r no. 1, the 5-bladed tanker p r o p e l l e r as no. 2, and the 6bladed p r o p e l l e r as p r o p e l -l e r no. 3. The container vesse-l p r o p e -l -l e r w i t h the largest blade area r a t i o is designated as p r o p e l l e r no. 4, the p r o p e l l e r w i t h the expanded blade area r a t i o of 1, 06 as p r o p e l l e r no. 5 and the p r o p e l l e r w i t h the blade area r a t i o of 0. 950 as p r o p e l l e r no. 6.
03 02
1.060
N. \ > / P 950
Figure 18. Blade contoux-s of designed propellers for third generation container vessel.
4. Characteristics of propellers in wakefields 4.1. Unsteady loads
4.1.1. Calculation program
F o r two w a k e f i e l d s , one typical f o r a s i n g l e -screw tanker and one of a t w i n - s c r e w container vessel, discussed i n chapter 2, two series of p r o p e l l e r s were designed according to the procedure given i n chapter 3. The f i r s t s e r i e s has been designed f o r the tanker and the p a r a -meter which has been v a r i e d independently i s the blade number ( z = 4 , 5, 6). Other p a r a -meters v a r i e d due to the underlying design procedure. The second s e r i e s , designed f o r the container vessel, has as independent parameter the m a r g i n against cavitation (Cp^.^/CT = 0. 7, 0. 85 and 1.0). Due to the design procedure the p r i n c i p a l dependent variable is now the blade area r a t i o .
In this section the results are given of the dynamic load on these p r o p e l l e r s , c a l c u l a t e d f o r the various w a k e f i e l d s . The calculations are p e r f o r m e d by means of the N . S . M . B . computer p r o g r a m m e based on unsteady l i n e a r i s e d l i f t i n g surface theory. The development of this p r o -g r a m m e was p e r f o r m e d by V e r b r u -g h [11], who adopted the f o r m u l a t i o n of the l i n e a r i s e d h y d r o -dynamical equations i n t e r m s of an acceleration potential given by Sparenberg [12], V e r b r u g h e x -tended the theory to include i n the computer p r o g r a m m e the calculation of the dynamic load on a p r o p e l l e r i n a n o n - u n i f o r m w a k e f i e l d . C h a r a c t e r i s t i c f o r the n u m e r i c a l procedure of the computer p r o g r a m m e i s the s t r a i g h t f o r w a r d treatment of the t r i p l e i n t e g r a l equation r e s u l t -i n g f -i - o m the t h e o r y . In an exhaust-ive way, r e g u l a r and singular parts of the integrands are separated. Regular p a r t s are integrated by p u r e l y
186
Table 3
Input data f o r p r o p e l l e r designs f o r typical large tanker w i t h conventional s t e m .
- i d e a l ( f r i c t i o n l e s s ) p o w e r a t p r o p e l l e r = 4 0 0 0 0 H P ( l H P = 7 5 k g f m / s e c ) , - p r o p e l l e r d i a m e t e r = 9 m e t e r , - p r o p e l l e r s p e e d = 90 r e v o l u t i o n s / m i n . , - s h i p s p e e d = 16 k n o t s , - e f f e c t i v e s t a t i c p r e s s u r e a t p r o p e l l e r s h a f t ( P ^ - P ^ ) = 2 7 0 0 0 k g f / m ^ , - r a k e a n g l e = 8 d e g r e e s , - p o s i t i v e m a r g i n a g a i n s t c a v i t a t i o n = 30 p e r c e n t ( k = 0 . 7 i n f o r m u l a 2 0 ) , - maximum a l l o w a b l e t e n s i l e s t r e s s 0,^, ( l e s s t h e t e n s i l e s t c a u s e d b y c e n t r i f u g a l f o r c e s ) i n k g f / m ^ f o r t h e v a r i o u s r a d i i i s as g i v e n b e l o w , - r a d i a l d i s t r i b u t i o n o f T a y l o r w a k e f r a c t i o n w ( x ) as d e t e r m i n e d f r o m F i g . 1 i s as g i v e n b e l o w , - T a y l o r wake f r a c t i o n , as d e t e r m i n e d f r o m w ( x ) - v a l u e s e q u a l s 0 . 4 5 9 , - p r o p e l l e r m a t e r i a l i s b r o n z e , - C„ = 0 . 0 1 a t a l l r a d i i . • X w(x) 0.2 0.250 4.0x10^ 0,3 0.265 3.6x10^ 0.4 0.280 3.2x10^ 0.5 0.300 2.8x10^ 0.6 0.335 2.4x10^ 0.7 0.405 2.0x10^ 0.8 0.495 1.8x10^ 0.9 0.585 1,6x10^ 1.0 0.680
-Table 4
P a r t i c u l a r s of designed p r o p e l l e r s f o r typical large tanker w i t h conventional s t e r n . 4-BLADED PRnPKI.LER FOR LARGE TANKER WITH CONVENTIONAL STERN
P r o p e l l e r No.1. p . = 4 0 0 0 0 HP; C T h i = 1.642; C p j = 1.619; 3 = 9 m; N = 9 0 RPM e = P - P O V = 27000 kgf/m^; T = 3 5 7 7 4 3 k g f ; = 0 . 4 0 1 ; 0 = 357764 kgfm; AE/AQ = 0.6 39; WEIGHT = 6 8 2 6 8 k g f "Pfj = 5 .953 m; PN/D = 0. 661; V = 8.230 m / s e c j X l - w ( x ) t a n S t a n 61 CL-C/D t [m] 1/Kc fo[tn] dT/dx dQ/dx 0.2 0.250 0.24? 0.844 0 0.482 2.299 0.454 0 4.772 0 0 O.J 0.265 0.171 0.571 0.171 0.457 2.619 0.710 0.159 5.527 175005 158628 0.280 0.156 0.454 0.151 0.591 2.876 0.896 0.101 5.558, 506064 255094 0.5 0. JOO 0.116 O.J53 0.128 0.541 5.070 0.905 0.068 5.599 450257 571288 0.6 0.3J5 0.108 0.505 0.108 0.281 5.201 0.841 0.082 5.701 541090 495200 0.7 0.«)5 0.112 0.272 0.093 0.216 5.269 0.759 0.081 5.951 65024J 643959 0.8 b.120 0.250 0.077 0.151 5.225 0.656 0.081 6.178 715820 771945 0.9 0.585 0.126 0.252 0.055 0.085 2.675 0.461 0.081 6.596 649569 765074 1.0 0.680 0.1J2 0.217 0 0.020 0,516 0.289 0 6.128 0 0
5-BLADED P R O P E L L E R FOR L A R G E TANKER WITH CONVENTIONAL STERN
P r o p e l l e r No.2.
P i = 4 0 0 0 0 HP; C T h j = 1.659 C p j = 1.619; D = 9 m; N = 90 RPM;
E = 8 ° ; PQ - Pv = 2 7 0 0 0 kg£/ra2; T = 3 6 0 6 4 1 k g f ; no = 0 . 3 9 9 ; Q = 3 6 1 9 5 4 kgfm;
AE/AQ = 0 . 7 0 0 ; WEIGHT = 7 1 9 0 3 k g f ; FN = 6.039 m; P N / D = 0 . 6 7 1 ; V = 8.230 m / s e c ;
x l - w ( x ) tan 6 tan 8j C L - C / D t[m] cCm] l / K c •fof™] P N M d T / d x dQ/dx
0.2 0.25 0.245 0.834 0 0.466 1.987 0.411 0 4.717 0 0 0.-5 0.265 0,171 0.564 0. 142 0.421 2.280 1.699 0.114 5.706 I8OI55 143030 0.-4 0.280 0.156 0.429 0.121 0.577 2.514 0.924 0.077 5-689 307808 255026 0.5 0.300 0.116 0.549 0.102 0.528 2.691 0.959 0.065 5.712 451406 571117 0.6 0,555 0.108 0.299 0,086 0.270 2.810 0.904 0.060 5.795 557955 492818 0.7 0,405 0.112 0.269 0.074 0.208 2.670 0.827 0.059 6.018 647799 644064 0.8 0,495 0.120 0.247 0.061 0.145 2.829 0.697 0.059 6.257 713789 777073 0.9 0.585 0.126 0.229 0.045 0.065 2.359 0.498 0.061 6.478 666244 787511 1.0 0.680 0.132 0.214 0 0.020 0.317 0.502 0 6.05a 0 0 6 -B L A D E D P R O P E L L E R F O R L A R G E T A N K E R W I T H C O N V E N T I O N A L S T E R N P r o p e l l e r No. 3. P J = 4 0 0 0 0 H P ; Cj.hj. = 1 .674 ; C p j = 1.619; D = 9 m; N = 90 R P M ; E = 8 ° ; p ^ - p ^ = 2 7 0 0 0 kgf/m2; T = 3 6 2 2 6 2 kgf; = 0 . 3 9 7 ; Q = 3 6 5 5 3 9 kgfm; AJ./AQ = 0 . 7 5 8 ; W E I G H T = 7 5 1 8 0 kgf; = 6 . H 8 m; F ^ / D = 0 . 6 8 0 ; V = 8.230 m/sec; X l - w ( x ) t a n 6 t a n S I CL-C / D t [ m ] c [ m ] 1/Kc Pf,[ra] d T / d x dQ/dx 0.2 0.25 0.245 0.826 0 0.452 1.763 0.594 0 4.675 0 0 0.5 0.265 0.171 0.559 0.122 0.409 2.058 0.679 0.098 5.885 186366 146863 0.4 0,280 0.156 0.425 0.100 0.365 2.259 0.939 0.062 5.797 305636 250541 0.5 0.500 0.116 0.346 0.085 O.3I8 2.424 1.017 0.051 5-810 430490 369846 0.6 0.375 0.108 0.296 0.070 0.261 £•554 0.976 0.045 5.870 530589 487691 0.7 0,405 0.112 0.266 0.061 0.201 2.590 0,893 0 . 0 4 4 6.093 645046 642877 0.8 0,495 0.120 0.245 0.051 0 . 1 4 1 2.550 0.750 0.045 6.524 708753 778239 0.9 0.5S5 0.126 0.227 0.058 0.080 2.148 0.535 0.049 6.559 685338 814618 1.0 0,680 0.132 0.212 0 0.020 0.517 0.526 0 6.002 0 0
188
Table 5
Input data f o r design of p r o p e l l e r s f o r t h i r d generation contamer vessel.
i d e a l ( f r i c t i o n l e s s ) power a t p r o p e l l e r =40000Hp ( l H P = 7 5 k g f m / s e c ) , p r o p e l l e r d i a m e t e r = 7 . 2 m e t e r , p r o p e l l e r s p e e d = 150 r e v o l u t i o n s / m i n , s h i p s p e e d = 30 k n o t s , e f f e c t i v e s t a t i c p r e s s u r e a t p r o p e l l e r s h a f t (.Pq-P^')^ 16000 k g f / m ^ , r a k e a n g l e = 8 d e g r e e s , p o s i t i v e m a r g i n s a g a i n s t c a v i t a t i o n = 0 % , 15% and 30% ( k = l , 0 . 8 5 and 0 . 7 0 i n f o r m u l a 2 0 ) , - maximum a l l o w a b l e t e n s i l e s t r e s s a l e s s t h e t e n s i l e s t r e s s 2 c a u s e d by c e n t r i f u g a l f o r c e s ) i n k g f / m f o r t h e v a r i o u s r a d i i i s a s g i v e n b e l o w , r a d i a l d i s t r i b u t i o n o f T a y l o r wake f r a c t i o n w ( x ) i s a s g i v e n b e l o w , T a y l o r wake f r a c t i o n , a s d e t e r m i n e d f r o m w ( x ) - v a l u e s e q u a l s 0 . 9 6 9 , p r o p e l l e r m a t e r i a l i s b r o n z e , Cj^=0.01 a t a l l r a d i i X w(x) CTrp 0.2 0.940 4.0x10^ 0.3 0.950 3.6x10^ 0.4 0.965 3.2x10^ 0.5 0.975 2.8x10® 0.6 0.975 2.4x10® 0.7 0.975 2.0x10® 0.8 0.975 1.8x10^ 0.9 0.970 1.6x10® 1.0 0.965
-Talïle 6
P a r t i c u l a r s of designed p r o p e l l e r s f o r t h i r d generation container vessel.
5 - B L A D E D P P 0 P E L L E 1 FOR FA.-iT CnNTAIMFR Ve.SS£h W I T H f p / A n = 1 . 2 1 0 P r o p e l l e r N o . 4 .
P I = 4 0 0 0 0 H P ; C T h T = 0 . 3 4 8 C p = 0 . 3 8 4 ; D = 7 . 2 n ; N = 150 R P ( ' ; e = Po - P „ = 1 6 0 0 0 k g f / m 2 ; T = 1 5 5 7 3 3 k g f ; Ho = 0 . 5 1 8 : 0 = 2 8 5 9 3 6 k g f m ; A p / A Q = 1 . 2 1 0 ; WEIGHT = 4 2 9 2 1 k g £ ; F f , = 6 . 9 2 3 n ; 7,^/0 = n . 9 6 2 ; V = 1 5 . 4 32 m / s e c ;
X 1 •w(x) tan 3 tan 6 j C L . C / D t[r,] c[n] 1 / K c £oM d T / d x d Q / d x
0 . 2 0 9 4 0 1 . 2 8 2 1 . 4 9 2 0 0 . 3 3 4 1 . 4 2 5 0 . 2 5 4 0 6 . 7 4 9 0 0 0 . 3 0 9 5 0 0 . 8 6 4 0 . 9 9 7 0 . 0 3 3 0 . 2 7 9 2 . 3 9 5 0 . 4 2 B 0 . 0 3 8 6 - 9 5 9 7 1 7 7 3 9 4 4 8 0 0 . 4 0 9 6 5 0 . 6 5 8 0 . 7 5 1 0 . 0 3 5 0 . 2 2 5 3 . 1 7 0 0 . 5 6 7 0 . 0 3 3 6 . 9 5 5 1 2 7 2 9 2 1 7 6 8 6 0 0 . 5 p . 9 7 5 0 . 5 3 2 0 . 6 0 2 0 . 0 3 4 0 . 1 7 6 3 . 7 4 9 0 . 6 0 3 0 . 0 3 0 6 . 9 5 2 1 8 1 8 1 6 2 7 2 8 9 2 0 . 6 0 9 7 5 0 . 4 4 3 0 . 5 0 2 0 . 0 3 1 0 . 1 4 0 4 . 132 0 . 5 8 8 0. 0 2 9 6 . 9 4 1 2 3 5 1 2 2 3 8 3 9 8 2 0 . 7 0 9 7 5 0 . 3 8 0 0 . 4 3 0 0. 0 2 8 0 . 110 4 . 3 2 0 0 . 5 5 7 0 . 0 2 8 6 . 9 3 4 2 8 3 1 2 6 5 0 6 4 1 5 0 . 8 a. 9 7 5 0 . 3 3 3 0 . 376 0 . 0 2 4 0 . 0 8 0 4 . 2 8 6 0 . 4 7 9 0 . 0 2 8 6 . 9 2 8 3 1 3 6 1 8 6 2 1 7 3 1 0 . 9 0 . 9 7 0 p . 2 9 4 0 . 3 3 4 0 . 0 1 8 0 . 0 5 0 3 . 6 3 8 0 . 3 3 7 0 . 0 3 0 6 . 9 1 6 2 9 6 5 5 3 6 4 9 6 1 1 1 . 0 0 . 9 6 5 0 . 2 6 3 0 . 3 0 0 0 0 . 0 2 0 1 . 0 6 1 0. 1 9 8 0 6 . 7 9 4 0 0
5 - B L A D E D PROPELLER FOR FAST CONTAINER VE.SSEL WITH A ^ / A Q = 1 . 0 6 0 P r o p e l l e r N o . 5 . 4 0 0 0 0 H p ; <^Thi 0 . 3 4 8 ; C p . j - 0 . 3 8 4 ; D - 7 . 2 IT); N = 150 R P r ' ; c = 8 ° ; Po - Pv - 1 6 0 0 0 k g f / m 2 , T = 1 5 7 9 0 1 k g f ; n o = 0 ' 5 5 3 ; Q = 2 7 1 9 7 5 k g f m ; A E / A O = 1 . 0 6 0 WEIGHT = 3 9 9 3 4 k g f ; Pf) = f . 9 4 1 ; P„ / D = 0 . 9 6 4 ; V = 1 5 . 4 3 2 m / s e c ; X l - w ( x ) t a n B t a n f J C L . C / D t [ m ] c [ m ] 1 / K c toC'"] P N [ " ] d T / d x d n / d x 0 . 2 0 . 9 4 0 1 2 8 2 1 . 4 9 2 0 0 . 3 3 4 1 . 4 2 5 0 . 2 9 2 0 6 . 7 4 9 0 0 0 . 3 0 . 9 5 0 0 8 6 4 0 . 9 9 7 0 . 0 3 3 0 , 2 8 3 2 . 2 0 8 0 . 4 0 3 Q . n 3 3 6 . 9 7 6 7 2 3 9 7 9 3 8 0 5 0 . 4 0 . 9 6 5 0 6 5 8 0 . 7 5 1 0 . 0 3 5 0 . 2 3 3 2 . 8 3 0 0 . 6 5 1 0 . 0 2 8 6 . 9 7 4 1 2 8 6 9 5 1 7 4 1 7 0 0 . 5 0 . 9 7 5 0 5 3 2 0 . 6 0 2 0 . 0 3 4 0 . 1 8 7 3 . 2 9 2 0 . 6 8 2 0. 0 2 6 6 . 9 7 2 1 8 4 0 8 3 2 6 6 4 7 0 . 6 0 . 9 7 5 0 4 4 3 0 5 0 2 0 . 0 3 1 0 . 1 5 0 3 . 5 9 4 0 . 6 5 5 0 . 0 2 6 6 . 9 6 0 2 3 8 2 4 1 3 7 0 5 5 7 0 . 7 0 . 9 7 5 0 3 8 0 ft 4 3 0 0 . 0 2 8 0 . 1 1 8 3 . 7 3 6 0 . 6 1 3 0 . 0 2 5 6 . 9 5 4 2 8 7 0 1 1 4 8 3 6 6 8 O . S 0 . 9 7 5 0 3 3 3 0 376 0 . 0 2 4 0 . 0 8 5 3 . 6 8 6 0 . 5 2 7 0 . 0 2 5 6 . 9 4 8 3 1 8 3 3 1 5 6 7 2 0 6 0 . 9 0 . 9 7 0 0 2 9 4 0 334 0 . 0 1 8 0 . 0 5 3 3 . 0 9 6 0 . 3 7 6 " 0 . 0 2 7 6 . 9 3 6 3 0 1 0 9 6 6 0 5 5 6 6 1 . 0 0 . 9 6 5 0 2 6 3 0 3 0 0 0 0 . 0 2 0 0 . 8 7 5 0 . 2 2 5 0 6 . 7 9 4 0 0
5 - B L A D E D PROPELLER FOR FAST CONTAINER VE.S.SEL W I T H A E / A O = 0 . 9 5 0
P I = 4 0 0 0 0 H P : 0 . 3 4 8 ; P r o n e l l e r No C p j = 0 . 3 8 4 ; 6 0 - 7 . 2 m ; N = 1 5 0 R P H ; c = 8 ° : Po - Pv = 1 6 0 0 0 k g f / m 2 : 1 = 1 5 9 4 6 6 k g f ; n o ' 0. 5 7 9 ; 0 - 2 6 1 9 4 4 k g f r ï ; A E / A Q = 0. 9 5 0 ; WEIGHT = 3 7 7 2 7 k g f : F N =• 6 . 9 5 7 : P N / D = 0 . 9 6 6 ; \ = 1 5 . 4 3 2 m / s e g X 1 - w l x ) t a n e t a n ei C L . C / D t [ m ] c [ n ] 1 / K c foM P NM d T / d x d Q / d x 0 . 2 0. 9 4 0 1 . 2 8 2 1 4 9 2 0 0 . 334 1 . 4 2 5 0 3 2 5 0 6 . 7 4 9 0 0 0 . 3 0. 9 5 0 0. 8 6 4 0 9 9 7 0. 0 3 3 0. 2 8 7 2 . 0 7 0 0 5 4 7 0 . 0 2 9 6 . 9 9 0 7 2 8 5 3 9 3 3 1 1 0 . 4 0. 9 6 5 0. 6 5 8 0 7 5 1 0 . 0 3 5 0. 2 4 1 2 . 5 8 2 0 7 1 5 0 . 0 2 5 6 . 9 9 2 1 2 9 7 2 0 1 7 2 2 0 5 0 . 5 0. 9 7 5 0. 5 3 2 0 . 6 0 2 0. 0 3 4 0 . 1 9 7 2 . 9 6 0 0 7 4 2 0 . 0 2 4 6 , 9 9 0 1 8 5 7 3 5 2 6 1 1 7 9 0 . 6 0. 9 7 5 0. 4 4 3 0 5 0 2 0. 0 3 1 0. 1 5 9 3 . 2 0 4 0 7 0 6 0 . 0 2 4 6 . 9 7 9 2 4 0 5 0 7 3 6 0 8 0 6 0 . 7 0. 9 7 5 0. 3 8 0 0 . 4 3 0 0. 0 2 8 0 . 125 3 . 3 1 4 0 6 5 6 0 . 0 2 3 6 . 9 7 2 2 8 9 8 1 9 4 6 7 2 1 8 0 . 8 0. 9 7 5 0 . 3 3 3 0 . 376 0. 0 2 4 0. 0 9 O 3 . 2 5 4 0 5 6 5 0 . 0 2 4 6 . 9 6 6 3 2 1 5 8 0 5 6 2 3 4 2 0 . 9 0. 9 7 0 0. 2 9 4 0 . 334 0. 0 1 8 0 . 0 5 5 3 . 7 0 9 0 4 10 0 . 0 2 5 6 , 9 5 5 3 0 4 3 4 1 5 7 4 1 0 2 1 0 0. 9 6 5 0. 2 6 3 0 . 300 0 0. 0 2 0 0. 7 4 5 0 2 5 0 0 6 . 794 0 0
R e v i s e d «'igures ' b e l o n E i n s
t o
I n f l u e n c e o f //ake and P r o p e l l e r
L o a d i n g and C a v i t a t i o n
b y;7. v a n Gent
and
P. v a n O o s s a n e n
•0 . 1 0
-0.6 ^ 0.8
A
1.0 0.2
0.6 ^ 0.8
191
numerical methods, while singular parts are integrated analytically. No additional assump-tions are made and thus i t can be assumed that the computer programme gives an exact s o l u tion. The f i r s t p r e l i m i n a r y results of the c a l -culation programme were reported by Kuiper [13]. F o r comparison w i t h the s i m i l a r computer programme of Tsakonas and Jacobs [14] the w o r k of Schwanecke and Laudan [15] i s of i n -terest, i n which a case i s considered where only the inhomogenity of the axial velocity component of the wake i s taken into account. The N . S. M . B . programme has also been extended to include the calculation of the dynamic load on a v i b r a t i n g r i g i d p r o p e l l e r . F o r this case a comparison w i t h experimental results o f W e r e l d s m a i s given by Van Gent [ 1 6 ] .
The physical meaning of l i f t i n g surface theory as applied to a p r o p e l l e r i n a nominal w a k e f i e l d can be explained shortly as f o l l o w s . Due to the l i n e a r i s e d approach the thickness and camber of the blades are l e f t out of consideration as they do not contribute to the dynamic load. The
only property of the propeller responsible f o r its hydrodynamical a c t i o n i s that of a f i n i t e n u m -ber of equidistant impenetrable bounded l i f t i n g surfaces.
The inhomogenity of the w a k e f i e l d causes a kinematical disturbance of the f l o w surrounding these l i f t i n g surfaces. The velocity component of the wake n o r m a l to these surfaces must be cancelled by a reaction of the load on the l i f t i n g surfaces. When the axial and tangential velocity components are taken into account the n o r m a l component i s : V^cosy + V^siny. The value of this function varies over the blades and i t is this disturbance function which appears i n the i n -t e g r a l equa-tion of -the l i f -t i n g surface -theory. This i n t e g r a l equation is i n f a c t the exact f o r mulation of the abovementioned boundary c o n -d i t i o n , which states that the velocity can only be tangential to the l i f t i n g surface. The solution of this equation gives the dynamic load.
FORCE RATIOS / i
Figure 20. Blade and Propeller loads for three propel-lers in the average tanker wakefield.
Q2 Q4 Q6 ^ 0 8 1.0 02 0 4 Q6 ^ Q8 10 Figure 21. Radial load distributions for five-bladed pro-peller for three different tanker wakefields.
F O R C E R A T I O
O l ?
Figure 22. Radial load distributions for three propellers in container vessel wakefield no. 6.
4.1.2. Presentation of results
The results of the dynamic load calculations are presented i n F i g u r e s 19 to 24. F i g u r e 19 gives a comparison of the r a d i a l load d i s t r i b u -tions on "a blade of the p r o p e l l e r s w i t h blade numbers 4, 5 and 6, each designed f o r the average tanker f i e l d . The load d i s t r i b u t i o n s are presented as the amplitude r a t i o s of the axial dynamic f o r c e s per unit length and the a x i a l static f o r c e per blade F ^ , These r a t i o s are given f o r the harmonics m = 3 to 7 of the shaft frequency. Also the r a d i a l d i s t r i b u t i o n s of the phase angles are given. F i g u r e 22 gives the same comparison f o r the three f i v e - b l a d e d p r o p e l l e r designed f o r the container f i e l d , each w i t h a d i f f e r e n t blade area r a t i o .
In F i g u r e s 20a and 23a the r a t i o s of a x i a l dynamic blade f o r c e F ^ and a x i a l static blade f o r c e F A are given f o r the various harmonics of the shaft frequency. F i g u r e s 20b and 23b give the bending moments M B of these f o r c e s i n a
1.06 „ . „ 0 9 5 121 106 „ 0.95121 106 ^ _ Q 9 5
Figure 23. Blade and Propeller loads for threepropellers in container vessel wakefield no. 6.
r a d i a l plane. These moments are made d i m e n -sionless by dividing by the static torque per blade M ^ . In the F i g u r e s 20c, d, e and 23 c, d, e, the six components o f t h e total p r o p e l l e r load at blade frequency are given.
F i n a l l y F i g u r e s 21 and 24 give a comparison of the dynamic load f o r one p r o p e l l e r i n v a r i o u s wakefields.
When the r a d i a l load d i s t r i b u t i o n s are c o n -sidered, i t appears that the v a r i a t i o n s along the radius are i-ather large (Figures 19 and 2 2 ) . I t can be concluded that the d i s t r i b u t i o n of a m -plitudes and phase angles f o l l o w the s t r u c t u r e of the inhomogenity of the w a k e f i e l d . T h i s means that the r a d i a l dynamic load d i s t r i b u t i o n is r a t h e r sensitive to the inhomogenity and l i t t l e smoothing occurs due to the chordwise extent of the blades. Another conclusion can be that the r e s u l t i n g d i s t r i b u t i o n of dynamic stresses i n t h e blade m a t e r i a l w i l l be complicated.
Due to the v a r i a t i o n of the phase angles along the radius the physical averages per blade ( F i g u r e s 20 and 23) are much s m a l l e r than the pure averages of the amplitudes. So i t can be
193 0 , 2 4 r 0 2 0 -WAKEFIELD 5 WAKEFIELD 6 O 24 0.2 0 4 0 6 X 0 . 8 10 0 2 0 4 0 5 V 0,8 1.0 0 2 0.4 0 6 y Q 8 10
Figure 24. Radial load distributions for propeller no. 5 in two different container vessel wakefields.
concluded that i t is not allowable to judge the dynamic load d i s t r i b u t i o n on a blade on the base of i t s average. The blade f o r c e s of various h a r -monic order decrease i n general w i t h increasing order but not monotonically. This has i t s c o n -sequence f o r the total p r o p e l l e r load of which the six components are composed of blade loads of order equal to m u l t i p l e s of the blade number and these multiples plus o r minus one. The total dynamic p r o p e l l e r load f o r the various p r o p e l l e r s designed f o r the tanker wakefield d e c r e a s -es w i t h increasing blade number, except f o r the horizontal transverse f o r c e and moment ( F i g u r -es 20c, d, e).
4.1.3. Variation of blade area ratio and blade num-ber
When v a r y i n g the blade number as an indepen-dent variable i n the design of a screw p r o p e l l e r f o r a certain ship, the influence on the dynamic load is mainly t w o f o l d . In the f i r s t place the
blade area r a t i o of one blade and the steady load of one blade w i l l v a r y . Supposed that the blade area r a t i o and steady load f o r the screw as a whole r e m a i n constant, these entities f o r one blade w i l l decrease w i t h increasing blade n u m -ber. In the design o f t h e p r o p e l l e r series f o r the t a n k e r f i e l d the blade area r a t i o and the steady load f o r the screw as a whole are not exactly constant, but the variations are s u f f i c i e n t l y s m a l l to maintain the above-mentioned tenden-cies f o r one blade. In the design o f t h e p r o p e l l e r s e r i e s f o r the container vessel f i e l d the blade area r a t i o is v a r i e d i n nearly the same p r o p o r -tion as f o r the tanker s c r e w s , but w i t h constant blade number. When F i g u r e s 20a and 23a are compared i t can be concluded that i n the l a t t e r case the e f f e c t of blade area r a t i o is much l a r g e r than i n the f o r m e r case where obviously the e f f e c t i s reduced by the v a r i a t i o n of blade number.
195
the harmonic components of the w a k e f i e l d and the corresponding blade loads which contribute to the total dynamic propeller load.
F r o m the foregoing i t may be concluded that the effect of blade number v a r i a t i o n on dynamic p r o p e l l e r load is mainly due to the accompany-ing selection rule of pertinent wake harmonics.
4.1.4. Influence of deviations in wakefield
The question has s t i l l to be answered whether or not the typical f i e l d s discussed i n chapter 2 can indeed be considered as typical f o r a group of vessels. The typical tanker f i e l d is found by averaging the various harmonic components of f i v e wakefields. F o r one harmonic (m = 5) the e f f e c t of the deviations f r o m the average f o r the f i v e bladed propeller has been investigated. In Figures 21a and b the axial and tangential v e l o c -ity components of 2 of the 5 wakefields and of the average f i e l d are given. The corresponding dynamic f o r c e s are given i n Figures 21c and d. It appears that the deviations i n the dynamic f o r c e s are just as large as the deviations i n the w a k e f i e l d . Thus i t may be concluded that the choice of the average wakefield is v e r y c r i t i c a l seeing as how the dynamic f o r c e due to the average f i e l d is not at a l l the average of the dynamic f o r c e s .
In F i g u r e 24 the two wakefields of the c o n -tainer vessels considered are compared f o r three harmonics. Again the deviations i n the dynamic load are just as large as the deviations i n the w a k e f i e l d components themselves.
4.2. Cavitation characteristics 4.2.1. Method of calculation
The theory i n c o r p o r a t e d i n the computer p r o gramme f o r the calculation of the type and e x -tent of cavitation of a given p r o p e l l e r i n a s p e c i f i e d w a k e f i e l d can be divided into two p a r t s . 1. The determination of the pressure d i s t r i b u
-t i o n on -the p r o p e l l e r blades, and
2. The determination of the extent and type of cavitation f r o m the calculated pressure d i s -t r i b u -t i o n and r e l a -t i v e p r o p e l l e r - i n f l o w p r o p e r t i e s .
The determination o f t h e pressure d i s t r i b u t i o n on the blades is p e r f o r m e d by means of the L e r b s l i f t i n g line induction f a c t o r method f o r the spanwise loading and 2-dimensional wing
1 . 0 0 . 9 ¬ 0 . 8 0 . 7 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 1 . 0 0 9 0 . 8 0 . 7 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2
\ \
\ \
\ \
'•/'' \ \
\ \
\ \
1 . 0 0 . 9 0 . 8 0 , 7 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2\ \
\
\ \
\
\ \
'1' \ \\
\
i
I > 5
^
^
^
^
(
\ \
\
\ \
\
\ \
\
r / R \ \
\ \
\
1 8 0PORT
6Figure 26. Calculated cavitation patterns for the 3 tanker propellers inthe averaged wakefield for the vertical up-right blade position.
