ME D DELANDEN FRAN
STATENS SKEPPSPRÓVNINGSANSTALT
(PUBLICATIONS OF TIlE SWEDISH STATE SHIPBUILDING EXPERIMENTAL TANK)Nr64 GÖTEBORG 1968
ON THEORETICAL PREDICTIONS OF
CHARACTERISTICS AND CAVITATION
PROPERTIES OF PROPELLERS
BY
C..A. JOHNSSON
ScANDINAvIAN UNIVERSITY BOOKS
SCANDINAVIAN UNIVERSITY BOOKS Denmark: ruxso&i, Copenhagen
Noròay: NIVERSITETSE'OBLAGET, Oelo, Bergen Sweden AXADEMPOBIAQET-QUPEBTS, Göteborg SVN8KA B0EPÖBLAOET/NorstedtaBonniers, Stockholm
PRINTED IN SWEDEN BY
Synopsis
In the present report the status of the vortex theory of propèllers is briefly outlined and some examples given of the rate of success attained when applying calculation schemes of different degree of accuracy to the solution of different design and analysis problems.
First the different kin4s of hydrodynamic calculations to be carried out when designing a propeller are briefly outlined. Some examples are given of propeller designs and the results of some comparisons between calculated and measured values of the number of revs, and efficiency reported and discussed. Further a simple method fòr cal-culating the open water characteristics for arbitrary advance ratios is briefly described and discussed and the results of such calculations compared with experimental results.
The possibifities of predicting the relative rOtative efficiency by calculations is discussed and the results of some quasi-steady calcula-tions of thrust and torque variacalcula-tions are compared with experiment
Some of the factors governing the inception and extension of dif-ferent kinds of cavitation are discussed and the results of some approxi-mate calculations for predicting the inception cavitation numbers for bubble cavitation, sheet cavitation and tip vortex cavitation and the extension of sheet cavitation are compared with experiment.
Sorné of the conclusions drawn from the results are summarized at the end of the paper.
1. Introduction
In the last years considerable progress has been made in the vortex theory of propellers. The introduction of high speed electronic com-puters for performing the calculations has been very important for this progress as many of the approximations judged as necessary earlier could now be eliminated. In spite of this progress the com-plexity of the problems met with in connection with propeller
calcu-lations prevents the accurate solution of other than very special
problems. In the present report the status of the vortex theory is
briefly outlined and some examples are given of the rate of success atiained when applying schemes of different degree of accuracy to the solution of different design and analysis problems. The methods used for these calculations are mainly those in use at SSPA which are described briefly. Reference is however in many cases given to other methods.
0(x) G(x, ) 00(x, ) A' 'IT' R nD 2. List of Symbols a = radius of vortex dD CD - drag coefficient ldbV2 dL CL = lift coefficient !-ldbV2 o D = propeller diameter dD = drag of profile dL = lift of profile f = camber
f2djm camber of two-dimensional profile
= camber of propeller blade section from lifting surface calculations
r
G = non-dimensional circulation coefficient D VA
= non-dimensional bound circulation
= non-dimensional, local bound circulation = non-dimensiònal, local free circulation
induction factors of the induced velocities
= advance ratio of propeller
L1 = f2dim/f3dim ber correction factor
kj i3dim/i2dimC0t10 factor for ideal angle of attack
T
KT= pD4n - thrust coefficient KQ
= pD5n2 - torque coefficiènt
= length of blade section
n = number of revolutions
p = static pressure
po = static pressure in undiinurbed flow
PC = cavitation pressure
pv = vapour pressure
Pi = propeller pitch in ideal flow
q V2 = dynamic pressure
= dynamic pressure based on velocity VA
Q = torque
r, r' = radius of blade section
D
R
6
10.7R
= VvA2+ (o.7D)2 =REYNOLDS number of propeller
V
t = thickness of blade section
T =thrust
UA = induced axial velocity at the propeller UR = induced radial velocity at the propeller UT = induced tangential velocity at the propeller
= tangential velocity component
V = inflow velocity to propeller blade section including induced velocities
VA = advance velocity of propeller
= mean wake of a radius
w0 = mean wake over propeller disk r
X
B
Xh = non-dimensiónal hub radius
z = number of blades
a = angle of attack of blade section
ß = advance angle of blade section
ß. = hydrodynamic pitch angle at blade section
= hydrodynamic pitch angle at infinity downstream of propeller
r
= circulation= pressure drop at profile
LXQ = torque variation, single amplitude
= thrust variation, single amplitude
Va = half the velocity difference between the suction and pressure sides due
to angle of attack
= pitch angle deviation due to lifting surface effect
= propeller efficiency
= relative rotative efficiency
x = GOLDSTEIN function
O = blade position angle
O = 2Z cos ß/xD=angular coordinate
0L = angular coordinate of leading edge
= total chord length in terms of angular coordinate
u kinematic viscosity p = density
test cavitation number
p V2
2
Lp
-a- = - incipient cavitation number
i PVA2
a
3. Calculation of Open Water Characteristics
at the Design Point
The most general application of the vortex theory is for designing a propeller for a certain combination of thrust T (or power P), speed VA, number of revs, n and diameter D. This generally means that the propeller should be designed for a certain combination of J and KT, thereby having a certain blade form, thickness distribution, radial circulation distribution and so on. The hydrodynamie parts of these calculations are normally performed as follows:
Lifting line calculations. At these calculations the chord length of the blade sections is put to zero and the blades are replaced by lifting lines, the circulation of which is varying in radial direction. Constant circulation along a streamline being a condition, free helical vortices are shed backwards. The induced velocities at different field points emanating from these vortices can be calculated by using the law of BlOT-SAVAnT, if the strength of the vortices is known. The strength of the bound circulation and the magnitude of the induced velocities being known at the lifting lines, the propeller thrust and pitch can be calculated. The calculation procedure is one of trial and error which finally gives values of the pitch in ideal flow of the dif-f erent blade sections dif-for a certain thrust.
Lifting surface calculations. At these calculations each blade is replaced by a surface of bound vortices. The strength of the vortices is different at different points of the surface. The strength of the bound vortices being known, the strength of the free vortices shed backwards can be determined and the induced velocities from the two systems of vortices can be calculated at different points of the blades. Integration of the values of the velocities obtained at a suffi-ciènt number of points over the blade surface gives the camber of the mean lines of the different blade sections and values of the pitch of the blade sections, the latter being more accurate than the pitch distribution obtained with the lifting line calculations.
Calculations of the induced velocities emanating from the thick-ness distributions. By replacing the thickthick-ness distributions of the blade sections by representative combinations of sources and sinks the induced velocities can be calculated for different field points and thereby the influence on the pitch and camber determined.
Calculation of the influence of the viscosity. The influence of viscosity manifests itself in three ways. First the profile drag has to
8
be considered when transforming profile characteristics to propeller characteristics. Secondly, due to viscosity, the lift in real flow is different from that in ideal flow. Thirdly, due to the fact that the
boundary layer develops in different ways on the pressure and
suction sides of a profile the chordwise circulation distribution will differ from that in ideal flow.
The drag and lift values for profiles can be obtained from wind tunnel data. The influence of viscosity on circulation distribution can only be obtained by performing extensive boundary layer calculations or experimentally from measurements of the pressure distribution.
5) Calculatiòn of the influence of the hub on the induced velocities. In order not to violate the boundary condition at the hub, the latter has to be represented by combinations of sources and sinks and (or) vortices. A thorough discussion of this problem, which is generally not considered in the calculation schemes in use at present, can be found in [11].
Lifting Line Methods Used
The methods used at SSPA for performing lifting line calculations are the GOLDSTEIN x-method and an induction factor method. Both methods are applicable to ccmoderately loaded propellers" in the sense defined by LERBS [1,2].
GOLDSTEIN'S x-method is used with x-values of TACKMIIÇDJI, meaning zero circulation at the hub [3]. This method presumes a prescribed radial circulation distribution (optimum distribution) but can be used with reasonable accuracy also for other circulation distri-butions. At SSPA this method is used primarily for the design of wake adapted propellers for merchant ships for which an accurate calcula-tion of the inflow angles to the blade seccalcula-tions seems to be of minor importance, because of the predominant influence of the irregular wake field. Further, due to the simplicity of this method, it is suitable för integration with other types of calculations such as strength cal-culations, calculations for determining the cavitation margins and so on. A computer program where the x-method has been integrated in such a way has been in use at SSPA for several years.
The induction factor method in most general use is that of LERBS
[1, 2]. In [4] some features of this method were discussed and a
modified procedure proposed. This modified procedure is now in use at SSPA [5]. The main differences between this method and that ofFor overcoming the singularity in the expression
UA
iÇ'.
dG dx1 A
VA 2JXh
dx xx
where
uA=axial component of the induced velocity = axial induction factor
G =
= dimensionless circulationD VA
x=r/R
x' =r'/R
r = radius of the free vortex
r' =radius of the blade section at which the induced velocity
should be determinedwhen x' -.x, LEEBS introduces the new variables and q' and repre-sents the circulation distribution and the distribution of A in FouRIER series with sine terms and cosine terms respectively. Thereby he obtains two trigonometric integrals, the principal values of which are known from GLAUERT. In the method used at SSPA Eq. (1) is integrated directly by the use of SmrrsoN's Rule without introducing the variable ç, the circulation distribution G=F(x) being represented by an 8th degree polynomial. In the region around x=x' the function is approximated by rectangles which, because of the symmetry of the function, is equivalent to the use of a rest term like that proposed by STRSCHELETZKY [4]. At the hub and tip this no longer holds and the values of ItA! VA for X=Xh and x= i are obtained by extrapolation.
it is claimed in [4] that by avoiding the transformation to p the
accuracy is increased at the end points for circulation distributions differing from the optimum one. For an optimum circulation distri-bution on the other hand the proposed method gives undesirable undulations in the pitch curves because of the difficulties, of represent-ing an elliptic type of curve by a polynomial. In such a case, however, the u-method and the induction factor method of LERBS give the same result and the former may be preferred because of its simplicity. In [4] it was shown that the pitch angle of the vortices at in-finity fl would be more representative than the pitch angle at the lifting line when determining the induction factors A(ß, xix') and i(ß, x/x'). This procedure has been followed in the computer pro-gram in use at SSPA.
'o .4 12 0.8 N o 05 N q-0.4 02
NOTATION LNTINO LINE METHOD
INS lACTOR METHOD SSPA PROCEDURE
LER8S
H -METHOD
-02 03 04 05 06 07 08 09 IO
BLADE SECTION ,IR
Fig. 1. Pitch ratios in ideal flow obtained by different lifting line methods.
Loading case 2 of [6].
Pitch distributions obtained in one case by using the two different induction factor methods and the x-method are compared in Fig. 1. The results of the ,-method and LEEBS' induction factor method have been reproduced fröm [6] where also the necessary data for performing the calculations can be found.
Lifting Surface Methods Used
In the computer program in use at SSPA, which is based on lifting line calculations according to the u-method, the lifting surface effect is considered by the use of an approximate camber correction based
on -LUDWIEG and GLNZEL data [7]. In connection with the induction factor calculations however a more rigorous method is used, based on the PIEN approach, as outlined in [5, 8]. When this approach is used it is presumed that lifting line calculations have been carried out and thus the induced velocities u4 and u- calculated. The aim of the lifting surface calculations is the determination of the velocity
changes zu and LUT emanating from the bound and free vortices at the blades. These velocities are determined in a number of points
over the blade by the use of the law of BIOT-SAVART. The pitch angle
at a point (x, O) on the blade is determined from
1wi
UA LUA1WOVA
VA tan(ß+ß)
(x, 6) = (2) 1TX U LUrJ -
VA where1w
UA1w0
VA tanTX
UTJ
VAß.(x)=pitch angle for a blade section x from lifting line calculations
(ß+ß1)
(z, 9)=local pitch angle at a point z, O w= local wake factorw0 = mean wake factor
The ordinates of the meanline over a reference line with the angle fl.(x) relative to the base line are obtained as
roo
/(0) =
tan ¿\ß(x, O)xdO/cosß (3)J
This equation gives the camber of the profile in first hand. As the calculations show that for a certain chordwise circulation distribution the form of the mean line is almost exactly the same as that of a two-dimensional profile with the same circulation distribution, the values of the maximum camber are often related to each other in the following way:
kff2thm/!3dim (4)
where
k1 = camber correction factor
t3dim= camber of propeller blade section from lifting surface calcu-lations
/2dUfl=camber of corresponding two-dimensional profile
In most cases the distribution f(6) is not. syminetrical around the midpoint of the blade section and thus a pitch correction is obtained. The ideal angle of attack obtained from lifting surface calculations
t.0 t
e 12
differs, from that of the corresponding two-dimensional profile and a relation factor, k can be defined
/cai=ci3dim/xj2dim (5)
It turns out that for the combination symmetrical chordwise circu-lation distribution and unsymmetrical blade form relative to the
ge-nerator line one gets x2d=O but £Xi3thmO and thus k9=oo.
The equations for determining the induced velocities are in the SSPA-method essentially the same as those given in PIEN'S original paper [8]. In the SSPA scheme the radial distributions of blade length and circulation are however approximated by polynomials in contrast to PIEN's scheme where the blade was divided into strips. The radial and chordwise integrations are carried out by the use of SmIPsoN's Rule.
In two reports by CHENG another, more extensive refinement of PIEN's work is described [9, 10]. In the latter of these reports [10] also expressions for calculating the induced velocities from circulation distributions of arbitrary form are deduced.
In the SSPA scheme the original limitation to uniform chordwise distribution of the bound vortices stifi exists. For calculations for the NACA a=0.8 mean line the theoretical distribution is replaced by a uniform chordwise distribution between x/l=0 and x/l=0.9 as in-dicated in Fig. 2.
05 0.4 03 02 W O
O/STANCO F606 LEADING EDGE 1I
Fig. 2. Chordwise distribution of bound vortices used in lifting surface calculatious for a=O.8 mean lines.
08 NOTA TON MOAN L/NO Nj
81066ES NACA .O.8. ¡.540 SSAA t.08 '547 0.5 u o 09 0.2 O 08 a? as
25
¿0
t
o
o
NOTATION ¿JET/NG SURFACE METHOD PIEN BER WIN SOPA ECKHAROT-140ff DA/II
/
/
/
/
/
/
/
NOTATION L1FT7N0 SURFACE METHOD
BER WIN
PIEN
SOPA
04 0.7 04 0.9 LO
BLADE SECTION .fR
Fig. 3. Camber correction factors obtained by different lifting surface methods. Load-ing case 2 of [6] with symmetric blade form and NACA a= 0.8 mean line.
07 08 op 10
BLAOE SECTION ,.,/R
Fig. 4 Correction factors for ideal angle of attack obtained by different lifting surface methods. Loading case 2 of [6] with symmetric blade form and NACA a=0.8
mean line.
S
14
In Figs. 3 and 4 results from lifting surface calculations according to the SSPÄ scheme are compared with the corresponding results from the DTMB-scheme (PIEN and CHENU), results according to KERwn's lattice method [11] and approximate values from ECKFIARDT-MOR-GAN'S original paper [6]. The loading case is the same as in Fig. 1, the mean line NACA, a=O.8 and most of th results have been reproduced from [12].
From Fig. 3 can be seen that the cambers obtained with the more accurate lifting surface methods agree very well while the-approximate values accoding to ECKUARDT-MORGAN differ from the other values. The ideal angles of attack obtained with the three rigorous methods fOr the case of a=O.8 mean line are shown in Fig. 4. They agree reasonaby well with each other but differ appreciably from the two-dimensional value wich is used in more approximte schemes like that
of ECKRARDT-MOBGAIÇ.
Corrections Used for Blade Thickness Effect and Viscosity
The only method available at present for calculating the contribu-tion of the thickness distribucontribu-tion of the blade seccontribu-tions to the induced velocities is that of KERWIN [11, 13]. From KERWIN's results it is evident that this influence manifests itself primarily as a pitch
cor-rection, the influence on camber being very small. As no
pro-gram is available at present at SSPA for calculating this influence, a provisional design diagram has been made, based on values in [13] and on some unpublished values obtained from DTMB. The difference between the approximate values of SSPA and those of KERwIN is shown for one particular case in Fig. 5.In the more approximate scheme used at SSPA for routine calcula-tions no correction for blade thickness effect is made.
For the viscosity correction on lift two-dimensional wind tunnel data are used. This influence is considered by applying an additional angle of attack dependent in magnitude on the type of mean line used:
NACA a=1 mean line x= 1.61 CL
NACAa=O.8 mean line x=O.39 CL where CL= lift coefficient for blade section.
Relative Importance of Pitch Corrections from Lifting Surface and Thickness Effects
In the diagram in Fig. 5 values are given of the corrections of the pitch angles due to lifting surface and thickness effects. The loading case is the same as earlier. In the diagram a curve is included which marks one per cent pitch correction. Although the magnitude of these corrections varies considerably from one case to another the diagram shows that these effects are by no means negligible.
\
\\
NOTATION--.-..- CURVE CORRESPONDING TO ONE
PER COlIC PITCH CORP.
TWO DIMENSIONAL IDEAL ANGLE
OF ATTACJÇ
THREE. DIMENSIONAL IDEAL ANGLE
OF ATTACK jy ISSPA CALCI
CORP. ANGLE FOR THICKNESS EFFECT ACC. TO KIR WIN
CORP. ANGLE POR THICKNESS EFFECT SSPA CALC.
\
\
N
N
---.--02 0.3 04 as 06 07 08 0.9 10 BLADE SECTION 4 .71RFig. 5. Relative importance of pitch corrections from lifting surface and thickness
effects. Loading case 2of [6] with symmetric blade form and NACA a= 0.8 mean line.
Comparison Between Calculations and Experiments
In order to check the rigorous method some propellers, designed for homogeneous flow, were manufactured and open water tests carried out in the towing tank. As the lifting surface computer program was strictly applicable only to the mean line NACA a= 1, this type of mean Ene was used for the first three propellers. The agreement between the predicted and measured relations between T, K1]J2
and J at the design point was not very good for these propellers, as can be seen in Table I.
'.6 .4 1.2 I.0 0.4 02
16
TABLE I. Predicted and Measured Relation8 Between Thru8t and Number of Revs. Propellers with mean line NACA a= I.
')nM
J
n
-O PR DESIONED WITH APPROA. METHOD PR0.S. DESIGNED WITH RIGOROUS METHOD
e 0 00 .2.3 o, 0.5 . .v
,
0 .10 6II
. . 1.0 15 ADVANCE NUM8ER JFig. 6. Difference between ruimber of revs. at design point, as calculated and measured at open water tests. Propellers with mean line NACAa=0.8. Numbers refer to Table II.
Propeller No. P 1149 P 1148 P 1150
Design value J Design value Kr
Number of blades
Blade area ratio
Circulation distribution Blade form Measured value KTM /KTM
100__-. 1)%
\AT /Design value KTIJ2
Corresp. measured value M
100
(! i)
o 0.637 0.195 4 0.53 Non-optimum Symmetrical 0.16117.4
0.481 0.598 +6.5 0.892 0.137 5 0.75 Optimum Symmetrical 0.09927.6
0.172 0.856 +4.2 0.892 0.115 3 0.45 Optimum Symmetrical 0.098 15.1 0.145 0.88 +1.4For the optimum propellers the n-method was used for the calcula-tions.
From the Table it
is evident that the propellers are heavily
underpitched. From these tests the conclusion was drawn that, due to the influence of viscosity, the symmetrical chordwise load distri-bution corresponding to the theoretical values of NACA a= 1 mean line is not realized in real flow, a conclusion which is supported by the fact that the two-dimensional viscosity correction on lift is rather large for this type of mean line. This means that the viscosity correc-tion in three-dimensional flow might be different from that in two-dimensional flow due to lifting surface effect. In view of this the use of the a=0.8 mean line can be expected to be more successful, as the viscosity influence on lift is very small for this type of mean line. This turns out to be true as is evident from the diagram in Fig. 6. This diagram shows the percentage difference between calculated and measured number of revs, at the design value of J for a number of propellers having blade sections with mean lines NACA a=0.8. The measured results are those obtained at open water tests in the towing tank. The values of the discrepancy between calculation and experi-ment has been deduced in the way shown in Table I.
In Fig. 6 results are included obtained with propellers calculated with the rigorous method as well as propellers obtained with the more approximate scheme used for routine calculations. From the diagram
in Fig. 6 it is evident that the propellers obtained with the latter
method are generally somewhat underpitched. This is most likely due to the fact that pitch corrections emanating from thickness and lifting surface effects are not included in this scheme. The number of revs. obtained with the more rigorous method are however almost within ±1 per cent from the desired ones for J-values higher than about 0.45. For this latter value the results are far off in two cases. The reason for these two propellers being underpitched is most likely that the region of "heavily loaded" propellers is entered, in which case the influence of the contraction of the propeller race can no longer be neglected when calculating the induced velocities. The reason why good agreement was obtained in this region in one other case might be that the circulation distribution was different from that used in the two other cases, the latter having more load at root and tip (see Fig. 7). Of course other parameters than J and the circulation distri-bution define the region of "heavily loaded" propellers. In [14] dia-grams are given from which deviations like those shown in Fig. 6TABLE II. Important Data for Propellers Calculated with Rigorou8 Method. Mean Line NACA a 0.8; No. in diagrams i 2 3 4 5 6 7 8 9 10 11 12 13 PropeUer No. P 1256 P 1257 P 1258 P 1259 P 1260 P 1302 P 1303 P 1220 P 1221 P 1207 P 1222 P 1313 P 1198 Design J 0.637 0.637 0.637 0.483 0.483 1.136 1.136 0.585 0686 0.954 1.264 0.412 0.954 Design Kr 0.193 0.193 0.193 0.168 0.167 0.249 0.249 0.172 0.215 0.228 0.250 0.280 0.223 Numberof blades 4 4 4 5 5 5 5 5 5 5 5 5
Blade area ratio
0.53 0.53 0.53 0.61 0.61 0.58 0.45 0;65 065 0.65 0.65 0.64 0.85 Thickness distr.
NACA6Gmod O6mod 66mod
18
16
16
16
66mod O6mod 66mod 66mod
16 66mod Wake distr. const. const. var. const. var. var. var. eonst. const. const. const. const. var. Hub ratio Xh 0.167 0.167 0.167 0.167 0.167 0.24 0.24 0.2 0.2 0.2 0.2 0.193 0.22 Skew back No Yes Yes Yes Yes No No No No No No Yes No Thickn. £O,2R1D 0.038 0.038 0.038 0.04 0.04 0.028 0.028 0.034 0.034 0.034 0.034 0.047 0.034
0 05 0,6 07 08
8LAOO SCTçOç,
Fig. 7. Radial circulation ditributions for propellers calculated with rigorous method. Numbers refer to Table II.
can be calculated in the case of optimum circulation distribution. The deviations thus obtained for the cases shown in Fig. 6 are how-ever very small Thus an explanation of the discrepancy met with in these two cases is still missing.
The important data of the propellers calculated with the rigorous method are given in Table II. Further, the radial circulation distri-butions are shown in Fig. 7 and the pitch distridistri-butions in ideal flow in Fig. 8.
When calculating the open water efficiency for the model case, the drag coeffióient is normally assumed to have the value CD= 0.00 8 for
all blade sections. In Fig. 9 calculated values of the open water
efficiency are compared with values obtained in towing tank and cavitation tunnel whereby this value of CD has been used. In the. diagram only results obtained with the rigorous method are included as the propellers calculated with the approximate scheme were all wake adapted. Also the points 4 and 5 of Table II have been excluded
20 j. i. 2.4 0.6 O_4 0.2 o
rar
0.2 0.3 04 05 06 07 08 09 1.0 82402 22Cr/ON /RFig. 8. Radial pitch distribution in ideal flow. Numbers refer to Table II.
In Fig. 10 the drag coefficients corresponding to the measured values of propeller efficiency have been plotted with the REYNOLDS number, based on the inflow velocity at 0.7 R, as basis. Although the cavita-tion tunnel data are too much scattered, it is evident from the dia-gram that the drag coefficient is fairly constant (approx. 0.008) down
to R0.7 4.5 . 10 where a steep raise occurs. Such a tendency has been pointed out before, see for instance [15], but earlier calculations were more approximate than those of the present report. Accordingly the dispersion between the points seems to be smaller in the diagram
L
0 ,5 2 REYNOLDS NUMBER R,,07 (3 -o 0.020 0.0I5 0.010 t' N Q 0.005 13Fig. 9. Difference between calculated and measured values of open water efficiency
for propellers calculated with rigorous method.
'3
5 RESULTS FROM TOWING TANK A RESULTS PROM CAVITATION TUNNEL
10512 £8 £2 . S 545 89/06 £12
RESULTS PRO/I TOWING TANK £ RESULTS PRall CAVITATION TUNNEL
*12 i? £2, £ £3 8LA05 LINE flAM/N PLOW) 0105 2 .3 4 5 5 7 8 9 105 REYNOLDS NUMBER R,,07
Fig. 10. Drag coefficients for propeller blade sections, mean values over the blades.
.2
22
4 Calculation of Open Water Characteristics for
Arbitrary Advance Ratios
Method for the Calculations
A rigorous solution of the "inverse problem" i.e. the problem of finding the open water characteristics of a given propeller requires very extensive lifting surface calculations. A number of iterative processes must be run through as the radial as well as the chordwise circulation distribution is not known from the beginning Anyhow procedures for making such calculations are under development in
USA.
While waiting for a more accurate solution, a relatively simple method has been introduced at SSPA and applied to some cases with considerable success. A short description of this method will be given here.
The starting point is the equation for calculating the lift coefficient for a blade section, when using the -method:
4ITD
CL1= x sin
ß tan (ß-ß)z
where
GL=lif t coefficient for a blade section i = length of blade section
z =number of blades
=hydrodynamic pitch angle, induced velocities included ß =hydrodynamic pitch angle, induced velocities not included
If the propeller has been designed for shock-free entrance, the angle of attack is zero (or c) at the design J-value. For an arbitrary J-value the relation between angle of attack and hydrodynaniic pitch angle
is
where
suffix i denotes design J-value suffix 2 denotes arbitrary J-value If it is further assumed that
dGL
GL2=CL1
+
-doc
cos
[sin
CL1
X2
+cos (ß12ß2)+sin (ßjß2) cot ß1
X1
- tan (1ßi)
(P1ß2)+sin (P1_ß)]
X2
Based on Eq. (9) a computer program has been made at SSPA which, after an iterative procedure, gives values of 2' ß2, K2, KQ2 and O2 for different values of the advance ratio, J2.
The main problem when using this program is to estimate the
values of dCL ¡d-x for the different blade sections.' A reasonable ap-proximation seems to be to use the theoretical value 2r, corrected in the following way2ir
dCL/dcc= - = 2ir. k1
(10)k3
where k1= camber correction according to Eq. (4).
The introduction of (7) and (10) involves two assumptions:
a) The total chordwise circulation distribution is divided in two parts.
G=G12+AG2 (11)
tan (ßlßj)dCL/doc sin (ß1/32)cot
cos (ß1-2)
where
G12 = circulation emanating from the design camber of a prescribed type of mean line
iG2=additive circulation, emanating from angle of attack only.
C
then the following relation can be obtained:
d CL
CL1+
do 2 sin
(ß11x2) tan (ß-2ß2)
(8)
0L1
-
X1 sintan (fl1ß1)
Assuming 2 to be small i.e. sin 2=O2, cos a=l a second order
equation is obtained having the following principal value:
(9)
24
This assumption means that if a ôirculation distribution, corres-ponding to an angle of attack for a thin plate, is applied to a propeller blade, the resulting camber curve should be that of a flat plate. From a diagram in [10], reproduced in Fig. 11 of the present report, it is evident that this assumption is far from exact, but seems to apply reasonably well for the middle and inner parts of the blade.
b) Eq. (10) further implies that not only should the resulting cam-ber curve be that of a flat plate but the ratio of the angle of attack of the plate in three-dimensional flow to that in two-dimensional flow should be the same as for the camber of the corresponding mean lines. To find out the accuracy of this assumption this ratio has been read in Fig. 11 in two ways. Firstly a line connecting the endpoints has been used as a reference flat p1ate Secondly, to the angle of attack thus obtained, the effective angle of atiack has been added, which is due to the induced camber and which can be approximately calcu-lated by the use of a f ormuhi of GLAUERT [16]. The three-dimensional correction coefficients kf obtained in these ways are compared with the corresponding pure camber coefficient kf for NACA a=0.8 mean line in Fig. 12. Again the deviations are most pronounced at the tip.
Fig. 11. Comparison of camber distributions due to angle of attack with that of a. two-dimensional flat plate. Reproduced from Ciro [10].
---S
- .----S-- --S S - -S--. --S--- -S NOTATION-
-
BLADE SOCTIOX .0.6 BLADE SECTION .0,9 BLADE SECTION .0.3EXACT 2-0 PLAT l'LATE
-N.
N
al 02 0.3 0.4 05 0.6 07 08 09 IO
DISTANCE FRON LEADING (DOE /l
0.05 o 0.05 I Ojo r Q as r 0.25 0.3
o
I
/
f
/
NOTATION,
- CAMBER. NACO. a-0.6 MEAN LINE
ANGLE OF ATTACX ONLY ANGLE OF ATTOCR CORP. FOR ADO.
CAMBER DUE TO 3-DIM EFFECT
-/
/
f/
/
f 02 03 04 05 0.8 0.7 08 0.9 F.0 8LAO0 SECTION .7/6Fig. 12. Calculated values of the three-dimensional correction factor k1 for angle of attack. Calculations based on ordinates in Fig. 11.
Comparison Between Caiculations and Eaperiments
A computer program based on the method outlined above requires among the input data the hydrodynamic pitch angle ßi1 and the lift coefficient CL1 for the blade sections at the design point, obtained by calculations with the -method. If the propeller has been designed from a systematic series or by any other lifting line procedure than the c-method it must be replaced by an equivalent -method pro-peller, the best starting point normally being that the pitch distribu-tion should be the same.
Other important input data are the drag coefficients for the blade sections. For the model case the formula CD=O.008+l.70c2 (cc in radians) is normally used in the SSPA calculations.
A favourable test case for the method should be one for which the propeller has been designed according to the c-method. The results of the comparison for two such cases are shown in Fig. 13. The pro-peller P 1018 has been designed for the same load, blade form and circulation distribution as propeller No. 2 in Table II, i.e. a homo-geneous flow design. The propeller P 1043 is the corresponding wake-adapted propeller (same case as propeller 3 in Table II).
In view of the serious approximations in the method the agreement between calculated propeller characteristics and those obtained from
26 'lo tOK. 100K0 0.7- 7- 7
uiiiiii
IîutuI-upÌ_uI
t NOTATIONPIO/8.OPEN WATER TE STO P/0I8.THEORET/CAL CALCULATIONS P104.3. OREN WATER TESTS PIONS. THEORETICAL CALCULATIONS
0/ 02 0.3 0.4 0.5 0.6 07 06 05 /.0 /1
ADVANCE RATIO J
Fig. 13. Propeller open water characteristics. Comparison between calculations and experiments. For data of propellers see Table Ill.
experiments, as shown in Fig. 13, is surprisingly good. Thus the rather small differences between the characteristics of the two propellers obtained at the experiments could be predicted from the calculations with rather good accuracy.
The method has also been tested on propellers for which the design principle was not known but the agreement between calculations and experiment was of the same order as that shown in Fig. 13. It still remains to test the method over a large range of blade ratios and pitch ratios.
From what has hitherto been experienced the method can however be regarded as suitable for use for instance for calculating scale effects in connection with self propulsion tests. Further the angles of attack and lift coefficients for the different blade sections, necessary for the theoretical prediction of the cavitation properties of propellers at different advance ratios, can be obtained with. this program. The results of such predictions are presented in Section 7.
A further development of the procedure outlined above would be to use induction factors instead of x-values for the lifting line
calcula-0.6 - 6 - 6 0.5 - 5 - 5 0.4 - 4 - 4 0.3 - j - 3 0.2 - 2- 2 0- 0- 0
tions. An attempt to make a computer program based on induction factor approach was made at SSPA. The difficulties in getting the iteration process to converge were however too great to make the scheme working successfully.
5. Theoreticál Prediction of Thrust and Torque Variations
The problem of calculating the thrust and torque variations for a propeller in a wake field involves the determination of the loading distribution on propeller blades of known geometry operating under known instationary flow conditions which is a problem of great
complexity. This task has initiated several scientists to
developunsteady lifting surface theories of different degree of accuracy and rigorousness. In a series of papers by TSAKONAS et al. the results of a continuous increase in rigorousness can be studied. The work done up to 1966 has been summarized in [17] where also reference is
made to other work of this kind.
As the evaluation and programming of an unsteady lifting surface theory is a tedious and difficult work, many attempts have been made to use quasi-steady calculation methods, sometimesincluding a correction term for unsteadiness, taken from 2-dimensional wing theory (SEns' function). In some cases good agreement with experi-ments is reported when SEARS' function is included, see for instance [18]. In [17] and [19] good agreement is reported between quasi-steady calculations and results from unquasi-steady lifting surface theory without including SEns' function.
A further simplification is to use the open water characteristics and the harmonics for a representative blade section (0.711). Methods of calculation based on this approach can be found for instance in [20] and [21].
Method Used for the Present Calcukdions
As the method for predicting open water characteristics outlined in Section 4 can be easily adapted for the purpose of predicting thrust and torque variations in a wake field it might be of interest to see how quasi-steady calculations carried out according to this scheme compare with experiments.
It can be proved, see for instance [22], that the thrust and torque variations are determined only by those terms of the FounIEn-series representation of the wake field which are integer multiples of the
28
number of blades, the first order component being predominant in most cases. Accordingly, when using the procedure outlined in Sec-tion 4 for this purpose three computaSec-tions have to be made in each
case:
One computation using mean wake components of the wake. Drag coefficient CD=0.°°8+1.7c2. This gives TM and QM.
One computation with the harmonics of the diffèrent blade
sec-tions added to the mean wake components. CD=0. Gives T4+
and Q4+.
One computation with the harmonics of the different blade
sec-tions subtracted from the mean wake components. CD=O. Gives
T4-. and Q4-. Finally
2T
-
T4T4..
(12a) T TM 2zQ (12 b) QMwhere 2T and 2iQ are the peak to peak amplitudes of the harmonics analysed.
Comparison Between Calculations and Experiments
In [18] the results of measurements of the thrust and torque
varia-tions on a series of 4-bladed propellers of different blade area ratios
are published. Assuming that the propellers had constant pitch
(optimum circulation distribution, not wake-adapted) the
hydrodynam-ic pitch angles and lift
coefficients at the design advance ratio could be calculated. From a lifting surface calculation the values ka3=: 1/ic1 (see Eq. 10) could be calculated and finally by making a Fourier analysis of the wake field the thrust and torque variations could be determined in the way outlined above. In the case of the first harmonic of the thrust variations different possibilities of per-forming the calculations were investigated which are illustrated in Fig. 14.From the diagram in Fig. 14 it is evident that when the radial
variation in the longitudinal wake harmonic is considered, better agreement is obtained with experiment than when only the value of radius 0.7R is used for all radii. When SEARS' correction factor for
15 IO e 'O o
-NOTATION
EXPERIMENT. KROHN (IB J
CALCULATED. HVAR. AVAR, 4TCONSL0
CALCULATED. ..VAR. 4.VAR.
CALCULATED. jqCDNSE 4.CONST.4oJR
CALCULATED. NVAN 4.VAR. SEARS IUNCT INCL,
NOTATION
Fig. 15. Thrust and torque variations. Comparison between calculations and
experi-ments for four-bladed propellers.
0.55 065 0.75
BLADE AREA RAllO
Fig. 14. Thrust variations. Comparison between calculations and experiments for
four-bladed propellers. Different principles of calculation.
EXPEPIJMENT. KROHN (161
CALCULATIONS. .VAR. 4 .1RA. CONSID.
.0--035 0.45 055 0,65
BLADE AREA RATIO
30
unsteadiness is included in the calculation scheme the thrust varia-tions are underestimated which confirms the experiences reported in [19].
In the case investigated the tangential component of the wake was not measured. In a recent investigation by HUSE [23] this was how-ever the case and as the wake fields were very similar in these two cases the fourth harmonics taken from HUSE'S report have been used for investigating the influence of the tangential component on the results. It can be seen in Fig. 14 that much better agreement is obtained with the experiments if the tangential wake component is considered
at the calculations.
In the diagram in Fig. 15 the calculated values of the thrust and
torque variations are compared with experimental results. The
diagram shows that the agreement between calculations and experi-ment is reasonably good both for the thrust variations and the torque variations when the tangential component of the wake variations is considered at the calculations.Results of experiments and lifting surface calculations reported in [24] show that when the blade area ratio is increased beyond the value 0.75 a diminution of the thrust and torque variations occurs. The explanation f òr this is according to [24] that the reduced fre-quency is getting so high that the unsteady effects, which tend to decrease the amplitudes, begin to dominate over the quasi-steady effect. If this is true it should mean that quasi-steady calculations of the kind demonstrated here should not be used for propellers having blade area ratios larger than about 0.75.
Another important fact, which is pointed out in [17], is that the wake harmonics, which are the basis for the calculations, are normally measured. without propeller. Results of experiments with body of revolutions show that the pressure of the propeller can affect the wake harmonics considerably. Further no method has hitherto been developed by which the influence of cavitation on the thrust and torque variations is considered.
6. Theoretical Prediction of Relative Rotative Efficiency When a propeller is put in the wake field behind a ship the loading of the blades is varying with the blade position in a way which affects the mean values of thrust, torque and efficiency. This effect is gener-ally lumped together with the effect of the rudder, influence of the turbulence level etc. in the concept of relative rotative efficiency R
If Kr-identity between the open water curve and the self propulsion test results is used, the following expression is obtained:
?R
7?B_ KQ0 (13)
7o -'--QB
where suffix O means open water condition and suffix B means behind condition.
The recent refinements of the propeller calculation methods could be expected to make it possible to predict the part of q which is due to the change in propeller loading caused by the wake field. Such calculations have been carried out by
YAzAxr
[19] for a loading case corresponding to a cargo ship. The results of his calcula-tions have been reproduced in Fig. 16. The calculacalcula-tions have been made in two steps. First the change in efficiency due to the change in the radial distribution of the mean load has been calculated using the mean wake for the different blade sections as input. Secondly the influence of the different harmonics of the unsteady components of the wake has been included in the calculations. From Fig. 16 it isevident that the latter part is predominant and there is a striking
influence of K on the magnitude of
In order to try to verily the calculations of
YIAzAxI
some experi-mental results obtained at SSPA have been included in Fig. 16. The results are of three different kinds:Measurements in the towing tank with a cargo ship model. Measurements in the towing tank with a cargo ship model fitted with nets flush along the model.
Measurements in the cavitation tunnel. Propeller behind a dummy model of a cargo ship. Model fitted with flushmounted nets. The tendency of YJvr.&zAxI's calculations is verified by all the experimental results except those obtained in the towing tank with a model without nets. The different trend of these results might be
32 1.08 N
::
1.02 0.98 NOTA TIaN--'.-
o dCOMPUTED. ff/5TA770144RY THEORY, ¿'AMAZAKI REME FROM 119) COMPUTED. AVERAGE WAKE. YA/lAZAN!. REME. FROM 1/91 MEASURED. TOW/NO TANK. CARGO OH/P MODEL MEASURED. TOW/NO TANK. CARGO OH/P MODEL WITH NETS
MEASURED. CAVITATION TUNNEL, DIP/MY MODEL WITH NETS WITHOUT RUDDER MEASURED. CAy/TAT/ON fl/RE/EL. DUMMY MODEL WITH NETS WITH RUDDER COMPUTED AVERAGE WAKE. SOPA 86-METHOD, DUMMY MODEL COMPUTED, AVERAGE WAKE. SOPA 86-METHOD, CARGO OH/P MODKE COMPUTED. AVERAGE WAKE. SOPA PRO FACTOR ME77/OD. CARGO 3M/P MODEL
0
0.96
Fig. 16. Variation of relative rotative efficiency with KT. Comparison between cal-culations and experiments.
explained by laminar separation on the blades at the self propulsion tests, the reason for which should be the combination of low turbu-lence level and low REYNOLDS number
The part of the relative rotative efficiency which is due to the in-fluence of the mean wake can be calculated by using the x-method according to the scheme outlined in Section 4. Some results of this kind have been included in Fig. 16 together with the results of cal-culations with the induction factor method, described in Section 3. When carrying out the calculations with the induction factor method the circulation distribution had to be the same as that obtained with the u-method because of the difficulties to establish a convergent procedure for induction fäctor calculations of this kind. The differ-ence between the results obtained with the two methods is rather great, the induction factor results probably being the more realistic ones. With both methods results are obtained which have a trend different from that of the results of YAltzAxI
''
°d' 0.10 o,e THRUST CA COEFF7. Theoretical Prediction of the Cavitation Properties of
Pro-pellers in the Model Case
Theoretical prediction of the cavitation properties of a propeller at a certain loading case can be expected to be less successful than the prediction of the corresponding overall characteristics like thrust, efficiency, thrust and torque variations and so on. The main reasons for this are:
Inaccuracy of the prediction of local inflow angles and induced flow curvature can be of great importance in this connection. Due to the rotation of the flow the pressure around a blade section is not exactly the same as for the corresponding two-dimensional profile.
e) Deviations from the theoretical blade section form are of impor-tance in this connection.
The knowledge of the factors governing the inception of different types of cavitation is not sufficiènt.
Also if the prediction is limited to the model case, the cavitation pressure under certain test conditions is not known with sufficient accuracy
In the f óllowing th results of some attempts to predict the incep-tion and extension of different types of cavitaincep-tion in the model case will be presented. The predictions are based on an engineering type of approach. Some of the points mentioned above will however be discussed first.
Pressure Field Around the Propeuer
If a profile is used as a blade section for a propeller the pressure field around the propeller has to be considered when calculating the pressure distribution around the section. This distribution can be calculated according to BERNouuI's equation. If we use a coordinate system moving with the velocity VA, cur, the pressure equation runs as follows:
34
whêre
Po = static pressure in undisturbed flow VA = advance velocity of propller
=rotational speed of propeller
UA, ' uR=axlal, tangential and radial components of the induced
velocity at the propeller disc. p = static pressure at propeller Thus we get for the pressure difference
2 ,2 U,2 2UA 2irx UT
qA/
q=+72+J72+Vjv
In Fig. 17 the pressure difference (-i--J is shown for different
pro-\ qA /
peller sections for propeller No. i of Table III at different J-values. The induced velocities used are those obtained by lifting line calcula-tions and the radial component of the induced velocities has not been included.
From Fig. 17 it is evident that these pressures are rather small, compared with those obtained from two-dimensional aerofoil theory. Including the induced radial velocities in the calculations will further decrease the magnitude of the pressure differences around the pro-peller blades. Accordingly, this effect will not be considered in the calculations following below.
q I.0 08 0.6 2 O 07 06 07 08 8LAOE SECTION. J.04 (15)
Fig. 17. Calculated pressures at the propeUer disk at different advance ratios. Pro-peller No. i of Table Ill. Influence of radial component of the induced
velocities not, included.
04
03 OS
020
o-ls
alo
Sos
Coo 0/STANCE FRON LEADING EDGE /I
influence on the Pressure Distribution of Deviations of the Blade Section. Form from the Theoretical One
The problem of calculating the pressure distribution along the blade sections of the propellers dealt with here is a difficult one for the following reasons.
First the outer blade sections are so thin that the i/i-values are outside the scope of the tables of the NACA-data [25]. Further, the NACA 66 modified section, as proposed in [6] was used, which dif-fers from the original NACA 66 form. Finally, it was necessary to deviate further from the original profile form near the tips in order to get snfficient strength, particularly at the aft ends of the sections. Calculations of how the deviations mentioned affect the pressure distribution were performed by using a computer program published in [26] which is based on THE0D0RsEN's method for profiles having finite thickness. The results of these calculations show a very strong influence of the leading edge radius, while the differences between the different types of NACA 66 sections, which are limited to the after part of the sections, seem to be of minor importance. Some of the results obtained for the blade section x=O.95 of propeller No. i of Table III are shown in Fig. 18. These results confirm partly the
re--0 022 IFROM PROF TARI.!)
0IN8 03V 0073
o I I I
0.1 02 0.3 0.4 0.5 0.6 0.7 0.6
0/STANCE FRON LEADING LOGE /I
Fig. 18. Influence of local changes of profile shape on pressure ditribution Results of calculations for a symmetrical profile corresponding to a propeller blade section at
36
suits of experiments reported in [27] where propeller blades were modified in a similar way and an appreciable influence on the incep-tion cavitaincep-tion number was obtained. The difference between the eìperiments and calculations is that the calculations give a change in pressure along the whole profile, indicating an influence particu-larly on the inception of bubble cavitation. At the experiments how-ever the most predominant influence was obtained for inception of pressure side cavitation.
As the change between the NACA 66 and 66 mod. sections and the further modified profiles of the propellers is limited to the after part of the profiles it seems justified to use data on pressure distributions interpolated in or extrapolated from the original NACA data in all the cases treated here.
Differences in Cavitation Properties Between Different Propellers with Small or Moderate Differences in Radial Distribution of Pitch and Cctmber
In view of the strong influence of the leading edge radius on the pressure distribution along the blade profiles, pointed out in the last
paragraph, a relatively large uncertainty can be expected in the
observations made in the cavitation tunnel, especially when the pro-pellers are manufactured according to normal routine and no device like the one described in [28] is available. In addition to this un-certainty there is also the scatter in the visual observations of cavita-tion between different tests, which is relatively large at cavitacavita-tion inception tests. These facts should be kept in mind when trying to detect experimentally such things as the effect of introducing more accurate calculation methods, the relative importance of using wake-adapted propellers, the effect of modifications in blade form etc. on the cavitation properties of propellers designed for a certain load-ing case. There is a risk that in some cases the influence of such modi-fications is so small that it is obscured by the influence of local differ-ences in the shape of the blade sections due to manufacturing in-accuracy. With this in mind it seemed to be appropriate to use, for the verification of some theoretical and semiempirical prediction methods outlined below, a material available at SSPA for one loading case, which has been used for testing different propeller design methods. Such a comparison could be expected to give an experimental veri-fication of the results obtained with these prediction methods as well as an idea of the magnitude of the influence of some of the factors mentioned above on the cavitation properties.o 0J
The material for this loading case includes results obtained with eight propellers designed for the combination J= 0.637 and K 0.195,
the other important data being those of propellers 1-3 in Table II. The main design and test data for the different propellers are given in Table III together with. brief remarks on the design principles used. The radial pitch distributions are given in Fig. 19 and the distribu-tions of maximum camber in Fig. 20.
The propellers of Table III can be divided into three main groups. Propellers designed for homogeneous flow. Blades with skew back. Propellers 1-4.
Propeller designed for homogeneous flow. Blades without skew back. Propeller 5.
e) Wake adapted propellers (cargo ship wake). Blades with skew back. Propellers 6-8.
The results of the observations of the occurrence of different types of cavitation for these propellers can be found in Figs. 21-26 and 30-31 and will be discussed in connection with the results of some theoret-ical and semiempirtheoret-ical predictions of cavitation which will be de-scribed below.
Fig. 19. Radial distributions of pitch for the propellers of Table III, designed for
the saine loading case.
r ______
!1
-ti
NOTATION
NOI800EN(OIJS PLOW DESIGN SKEW OACK
HOMOGENEOUS PLOW DESIGN NO SKEW BACK
WAKE ADAPTED OfS/ON SKEW BACK
-02 03 04 85 06 87 0.8 09 1.0 80.80E SECTION 12 O eA 0.2
38 2.5 20 '.0 o
!IIÇI
NOTATIONH07OGF.NEOUS FLOW DESIGN SKEW 6ACK HOMOGENEOUS FLOW DESIGN NO SKEW
BAcIO
WAKE ADAPTED DESIGN SKEW BACK
-
-0, 0.2 0.3 06 0.7 06 09 '.0
BLAOt SECTION .,/5
Fig. 20. RadiaI distributions of camber for the propellers of Table III, designed for
the same loading case.
Prediction of the Inception of Bubble Cavitation in Homogeneous Flow The inception of bubble cavitation is determined by the pressure distribution in the middle part of the blade profiles. The assumption that the cavitation pressure is equal to the vapour pressure leads to the following condition
where
a= cavitation number for cavitation inception
(zp/q)=maximum pressure reduction along the profile.
In reality the cavitation pressure is different frOm vapour pressure which leads to pc_pv
ai=I--1
+
\q/max
q where cavitation pressure p= vapour pressureIt is well known that the cavitation pressure depends on water speed (or static pressure), air cpntent ratio, tunnel configuration etc. [29]. In the present case it was decided to determine the cavitation
TABLE III. Data/or propellers designed for the same loading case. (Loading case corresponding to propellers 1-3 in Table II.) No. in diagr. 1 2 3 4 5 6 7 Propeller No. P 1018 P 1040 P 1084 P 1257 P 1256 P 1043 P 1085 P 1258 Design KT 0.196 0.194 0.193 0.193 0.193 0.209 0.195 0.193 MeasuredKT 0.192 0.207 0.202 0.188 0.200 0.210 0.187 0.190 Skew bark Yes No Yes Wake distr. Constant
Variable, typical cargo
Design method I II III TV IV I III
Lifting line method
x-method
md. f. LEEBS
md. factor SSPA
x-method
Lid. factor SSPA
Lilting surface method
Approx. CINZEL Guu.i.0T0N PIEN-SSPA Approx. GINZEL GUILLOTON Pirn'î-SSPA
40
pressure from the test results obtained at the design advance ratio J=0.637 according to the following equation, transformed from (17):
pp0=q(a (p/q),)
(18)These values were calculated with the experimental results for the five propellers of Table III, designed for homogeneous flow, as basis. The mean of the values for these propellers at the blade sections x=0.6, 0.7 and 0.8 was used as a representative mean value, as the calculations of inflow angles and velocities could be expected to be niore accurate at the middle part of the blade. The mean value thus obtained was
pp,= 390 kp/m2
for a water speed of 4 rn/s. This value compares reasonably well with the cavitation pressures reported by SSPA in [29] where values of pcpv ofSO to-250 kp/m2 were obtained at water speeds between 5 and 11 rn/s.
The cavitation pressure being known, the inception cavitation
number for different blade sections can be determined from the
calculated pressure distributions along the blade sections which are
based on angles of attack from Eq.
(9). Such calculations werecarried out for the three groups of propellers of Table III for different advance numbers J The results are ôompared with experimental results in Fig. 21.
J. 0.8
0.6 0.7 08 0.9 1.0
BLADE SECTION .,/R
NOTATION
NOM056NEOUS FLOW DES/SN
CALC VAWES
NODIOSENZOUS FLOW DES/SN EXP VALUES
WAKE ADAPTED DES/SN CALC VALUES
WAKEADAPTEO 005/SN EXP
VALUES
0.7 - 0.8 09 10 05 - 0.6 0.7 0.8 0.9 ¿0
BLADE SECTION o.,IR BLADE SECTION .JR
Fig. 21. Incipient bubble cavitation in, homogeneous flow. Comparison betweeñ cal. culations.and experiments. Numbers refer to Table III.
From the diagram in Fig. 21 it is evident that the agreement be-tween calculated and measured values of incipient cavitation number for bubble cavitation on the suction side is good.
The results of the calculations indicate a small difference between the inception cavitation numbers for the propellers calculated for homogeneous flow and the wake adapted propellers. These differences seem to be within the experimental accuracy.
Prediction of the Inception of Sheet Cavitation in Homogeneous Flow The factors governing the inception of sheet cavitation are not known very well. It is however clear that an angle of attack of the profile is necessary if sheet cavitation shall be present. Accordingly
it is generally accepted that sheet cavitation is a type of vortex
cavitation. To be able to predict, at least approximately, when sheet cavitation occurs, it is necessary to get some quantitative information on the following subjects:distribution of vortices over the blade
pressure in the vortices i.e. a relation between the vortex strength and the pressure
cavitation pressure for this type of cavitation or any law governing the time necessary for cavitation inception following a decrease in pressure.
In the following some results will be reported of calculations fol-lowing these lines. The methods for the calculations are very
ap-proximate but nevertheless the comparisons reported here show
reasonable agreement between experiments and calculations in mostcases.
The problem of determining the distribution of vortices over the blade is met with at lifting surface calculatiOns. The majority of those
calculations are however performed for the design point, where
shock-free entrance is supposed to be a condition.Equations for the free vortex distribution in the case of an angle of attack have been derived by CHENG [1O (the bound vortices are negligible in this case and will not be treated here). In the expression for the free circulation on a lifting surface
d f'°L()
G9(x, O)
-
I G(x, O)dOdr (19)42
CNG uses the folldwing expression for the bound circulation 0(x) 2
G(x 9)=
cot-ir 2
where
09(x, 9) =nondimensional free circulation distribution on the lifting surface
0(x), G(x, 9)=nondiniensional bound circulation
9T= total chord length in terms of angular coordinate O=angular coordinate of leading edge
which gives
G(x, O)
[G'(x) (&- sin )+G(x)'(1+cos &)]
(21)with
2O'L(x)OT'(x) (1cos )
OT(x) sin ir
The values of the circulation at the leading edge G(x, OL) and G9(x, OL) which are those of interest in this connection will both be infinite when determined by Equations (20) and (21). Besides no attention is paid to the fact that there is a variation in shape of the curve representing the bound circulation for sections of different shape. It was therefore decided to use for our purpose an approxima-tion for G(x, OL) obtained by representing the part of the bound cir-culation curve near the leading edge by a 2nd degree polynomial.
A(OLO)2
B(OLO)
G(x, 9=
°T2
+
j1C (22)
After insertion in (19) this gives the following simple expressions for the circulation at the leading edge
G(x, OL)=C
00(x, OL)=C O'.L=G(x, °L) (23)
The. value of the bound circulation G(x, OL) can be obtained from profile tables through the relationship
21'(x, O)=2IXVa
(20)
2(OL(x) O)
= 4G92=4(G(x, OL)O'L)2 (25)
where 2tv0 is the velocity difference between the suction and pressure sides due to angle of attack. It was considered realistic to assume that one half of this difference is acting on the pressure side and the
other half on the suction side, also at the leading edge.
The next problem to be solved is that of calculating the pressure emanating from the circulation according to (23). In [4] an approxi-mate expression was given, based on the well known formula for the pressure p in the centre of a vortex
pF2
Pc=Pb 2a2 (24)
where
pb=pressure at the boundary of the vortex F = circulation round the vortex
a = radius of the vortex
The following approximation was used in [4]
F G0dr
a d'r/2
which after insertion in (24) and nondimensionalizing gives
PbPc
qA qA
Finally the problem of determining the cavitation pressure for this type of cavitation has to be solved. As a basis for these particular calculations the following formula mentioned by STRASBERG [30] was used for determining the conditions for inception.
RmaxbT( p)If 2 (26)
where
Rm=maximum radius of a bubble formed by the onset of vapour cavitation
b =a constant
1p =peak positive value of pp(t)
p(t) =pressure transient
T
=time that pp(t) is positive
As the cavitation inception is often defined as the pressure at
which the bubbles have reached a certäin size, Eq. (26) might be usèd in the present case.
44
In order to determine the constant b a comparison is made with
the conditions for bubble cavitation at the design point. We can
writeT1 jI(p-P)1
=T2T'(PoPmin)2where suffix 1 denotes design point and 2 an arbitrary advance ratio. Further
T
(x/l)l (x/l)l VA- VVAT
pvpflin=qA((piqA)max a)
where i = profile length(xfl)=part of profile where the pressure is negative
V =inflow velocity to profile
= inception cavitation number based on velocity VA Thus
(xli), (z Ji)n21/qA2( ( p/qA)max - cre)
c
y1 VqA1((1pJqA)maaj)i V2
The constant o can be determined from the results of the tests for determining bubble cavitation inception. Using the data mentioned earlier in connection with predictions of bubble cavitation the value c=O.07 is obtained.
It should be emphasized that using Eq. (27) with the value c=O.07 for predicting inception of sheet cavitation involves several serious approximations and assumptions. For instance in Eq. (27) two types of pressure distributions are compared which differ appreciably in shape. Further the value of e has been determined at a certain com-bination of water speed (4 mis) and static pressure in a certain tunnel configuration.
The results of an application of the method outlined above on the
propellers of Table III for two advance numbers J are shown in
Fig. 22, where also experimental results of incipient sheet cavitation are included.In Fig. 22 the results for J=O.4 refer to sheet cavitation on the suction side while for J=O.8 pressure side cavitation was studied.
The following conclusions can be drawn from Fig. 22:
The dispersion between the experimental results obtained with the different propellers is rather large. The. main reason for this is probably
0.6 0.7
f
i I,,;\
:
i /J
r
II
/
O 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 (.0 6 J-O.8 o 45Fig. 22. Incipient sheet cavitation in homogeneous flow. Comparison between
calcula-tions and experiments. Suction side cavitation at J=04 and pressure side cavitation at J=O.8. Numbers refer to Table III.
the sensitivity of this type of cavitation to the form of the leading edge. Further it is very difficult to get accurate experimental results
as the extent in radial direction of leading edge sheet cavitation
changes very rapidly with cavitation number. This applies particularly to pressure side cavitation.In view of this the agreement between the calculations and the experiments is reasonably good. Unfortunately the dispersion of the results prevents an experimental confirmation of the difference be-tween the inception cavitation numbers for blades with and without skew back which can be predicted by using Eq. (25). What regards suction side cavitation the difference between the homogeneous flow designs and the wake-adapted propellers is of the same order when predicted by calculation. or obtained from experiments. For pressure side cavitation the difference is however exaggerated by the calcula-tions.
70 NOTATION
HOMOGENEOUS PLOW DESIGN SKEW BACK CALC VALUES
HOMOGENEOUS PLOW DESIGN SKEW BACK EXP VALUES
-
HOMOGENEOUS PLOW DES/ON NO SKEW BACK CALC. IALUES60 1
--u--HOMOGENEOUS PLOW DESIGN NO SKEW BACK EXP VALUES
WAKE ADAPTED DESIGN SKEW BACK CALC VALUES
//
WAKE ADAPTED DESIGN SKEW BACK EXP VALUESI. J-O.4
/
¡50
BLADE SECTION A-,-/R BLADE SECTION BIIR
20
'O
o as
46 70 60 50 40 So 20
\
NOTATION\
.HOMOGENEOUS FLOW DESIGN SKEW 64CR CALC VALUES
HOMOGENEOUS FLOW DESIGN SKEW BACK EXP l9LUES
HOMOGENEOUS FLOW DESIGN NO SKEW BACK CBLC VALUES HOMOGENEOUS FLOW DESIGN NO SKEW BACK EXP. VALUES
WAKE AOAPTEO OES/GB SKEW BACK CALC. VALUES
WAKE ADAPTED DESIGN 511(16 BACK EXP VALUES
-- -- ----
.
-\
\ \
,'N N.
N. N
"N N,\
---. -.-. -S..
. . 1-. -.
40 0.45 050 0,55 0.60 0.65 ADVANCE RATIO 'JFig. 23. Incipient tip vortex cavitation in homogeneous flow. Comparison between calculations and experiments. Numbers refer to Table III.
Prediction of Incèption of Tip Vortex Cavitation in Homogeneous Flow With the very approximate methods used here the only possibility to predict the inception cavitation number for tip vortex cavitation at a certain advance ratio seems to be to use values extrapolated from the predicted values of sheet' cavitation for different radii. It is
evident from the diagram in Fig. 23 that reasonable results are ob-tained with this approach. It is indicated by the experiments as well as by the calculations that the influence of skew back on the inception of tip vortex cavitation is very small while the difference between the homogeneous flow designs and the wake adapted propellers is appre-ciable. The importance of the design method for this type of cavita-tion is ifiustrated by the diagram in Fig. 23 where propellers designed with more rigorous calculation methods (higher numbers in each group) show lower values of a1.
