The Wageningen B-Screw Series

Transcription

1 The Wageningen B-Screw Series By W. P. A. van Lammeren] Member, J. D. van Manen/ and M. W. C. Oosterveld, 2 Member Member, The Wageningen B-screw series have been extended gradually to 21 screw series having blade numbers ranging from 2 to 7 and blade-area ratios between.3 and 1.5. Recently, the existing screw series were correlated with the new screw series with an upto-date fairing technique. This correlation was made by means of a regression analysis. Further, the influence of the Reynolds number (scale effect) on the test results was taken into account. For ship maneuvering studies, it is necessary to know the propeller thrust and torque characteristics over a wider range of operating conditions. Therefore, openwater tests with B-series screws over the entire region of operation were: carried out. n order to obtain systematic knowledge on the cavitation characteristics of the B-screw series in a uniform flow, tests have been conducted. The curves for cavitation inception were established and the influence of cavitation on propeller thrust and torque was determined. ntroduction AN MPORTANT method of screw design is that based on the results of open-water tests with systematically varied series of screw models. These screw series comprise models whose characteristic dimensions, such as pitch ratio P/D, number of blades Z, blade-area ratio A u/ao, blade outline, shape of blade sections, and blade thickness are systematically varied. Among the well-known screw series as developed by Schaffran, Taylor, Gawn and others, the Wageningen B-screw series of the Netherlands Ship Model Basin take an important place. The B-series screw type is frequently used in practice and possesses satisfactory efficiency as well as reasonable cavitation properties. The first tests with systematic series of screw propellers were performed at the Netherlands Ship Model Basin in From model experi- Directors, Netherlands Ship Model Basin, Wageningen, The Netherlands. =Head of Research Department, Netherlands Ship Model Basin, Wageningen, The Netherlands. Presented at the Annual Meeting, New York, N.Y., November 12-14, 1969, of THE SOCETY OF NAVAL ARCH- TECTS AND MARNE ]~NGNEERS. merits carried out by Baker and Riddle [1]a and Baker [2] it had become evident that screws used so far which had circular-back blade sections and elliptical blade outline (Taylor and Sehaffran) were in many cases inferior to screws with airfoil sections. Tests performed at the Netherlands Ship Model Basin confirmed these conclusions in many cases. Based on these results a series of four-bladed model screws of a shape similar to Baker's was manufactured and tested in the open condition. This series was called the A 4:-4 series (thus 4-bladed series with.4 blade area ratio). The results of the open-water tests with this series were given by Troost [3]. Later on it was found that the A 4-4 series screws, owing to the narrow blade tips and the airfoil-shaped sections over the whole blades, were only suitable for use in cases where no cavitation danger was present. n addition, the A 4-4 series screw type did have tmfavorable backing characteristics. These con,dderations led to the design of screws with wider blade tips, circularback seetions near the blade tips, and airfoilshaped seetions near the hub. Screws of this type a Numbers in brackets designate References at end of paper. 269

2 Table 1 Summary of the Wageningen B-Screw Series Btade number Z Btade area ratio A~/Ao 2 o.3o 3.55 # _.55 o.6o oo were called B-series screws. n this way the B 4-4 screw series was designed and gradually extended to further series with larger blade-area ratios and various blade numbers. These extensions were partly (mainly concerning the higher number of blades and the larger blade-area ratios) sponsored by the (former) Bureau of Ships of the United States Navy. The results of the openwater tests were given in a number of publications by Troost [3], and others [4], [5], and [6]. At present, about 12 screw models of the B- series screw type have been tested at the Netherlands Ship Model Basin. Table 1 gives a summary of the series. n general, the results of the tests were given in the form of KT and Ko coefficients expressed as a function of the advance coefficient d r for analytical work, and in the form of Bp-6 and Bu-6 diagrams for design purposes. From a correlation between the available diagrams of the B-screw series it appears that small differences exist. However, during the last years the B-series have been extended considerably and a cross-fairing of the B-screw series diagrams for different blade-area ratios and probably for different numbers of blades must be possible now. Recently we have started the fairing of the B- screw series test results by means of a regression analysis. As a result of this analysis, the thrust and torque coefficients KT and Ko of the B-series will be expressed as polynomials of the advance ratio J, the pitch ratio P/D, the blade area ratio AR/Ao, and the number of blades Z. n addition, the effect of Reynolds number on the test results was taken into account by using the method derived by Lerbs [7] from similar methods used for the calculating of the performante characteristics of airscrews from the characteristics of equivalent blade sections. n the future, the effect of Reynolds number may be taken into account in the polynomials as well. By = loading coefficient, NP ~/~ B,- VA % c = chord length of blade section CD = drag coefficient Cf = skin-friction drag coefficient CL = lift coefficient CT = thrust coefficient, T CT /2pVA2 --4 O n,n = Cr* = thrust coefficient, CT* = T 7r ~/2p[Va 2 -t- (.77rnD) 2] ~ D 2 Co* = torque coefficient, Q Co*= V..p[VA 2 + (.77rnD)q -~ D~D 4 Nomendature d = hub diameter D = propeller diameter J = advance coefficient, J - K~, = thrust coefficient, K o = torque coefficient, VA nd T KT-- on2d ~ O Ko = - - pn2d 5 number of revolutions per second and per minute p = pressure pv = vapor pressure p~ = static pressure of undisturbed streanl _P = power O = torque r = radius R = propeller radius Re = Reynolds number 27 The Wageningen B-Screw Series t = maximum thickness of blade section T = thrust V, = undisturbed stream velocity Z = number of screw blades AE/Ao = blade area ratio of screw.p/d = pitch ratio of screw d/d = hub diameter ratio = angle of attack of blade section /3 = hydrodynamic pitch angle at.7 R, /3 = aretan VA/O.7rrnD = speed ratio, ~ = 11.27/J p = specific mass of water ao = cavitation number, p -- pv ~/2. VA 2 no = open-water efficiency, J Kr 2~ KQ = kinematic viscosity of water

3 ~R Pitch dlstibution, _ L Ji 1 --~-4x! \! \! ~ ~ 4 = "', o~ \ 2R --t--n---- ~. _--4---~ / '~ _ J-- --T ----'~./ \ q ~/ ~ 822 /,,~ s~-~o,~'-- r "-1, -- ', " -7 T--~=--~-- ~:;,-.,,o ' 2' ~so'~.: B z-5 = B 4-7 ~ 4-65 u ~.-nxu Fig. 1 General plan of B 4 screw series o ~ l : &~ '~! ", ( =:~ Pitch distribution 2R ~_k ' "m >/ [ c31!x B ~22 ~-2N~u L Fig. 2 General plan of B 5 screw series The polynomials and the corrected design diagrams of the four- and five-bladed B-series screws are given in this paper. n order to obtain data for analyzing the manenvers of ships, submersibles, drilling vessels, and so on, open-water tests with a part of the B- screw series were conducted over a wider range of advance coefficients. This experimental program covered the following items: speed ahead, rpm ahead speed ahead, rpm astern speed astern, rpm astern speed astern, rpm ahead (lst quadrant) (2nd quadrant) (3rd quadrant) (4th quadrant) The effect of pitch ratio P/D, blade area ratio A E/A o, and number of blades Z on the characteristics of the B-screw series in the four quadrants have been determined. n addition, the results of the four-quadrant measurements are analyzed in the form of a Fourier series and the Fourier coefficients are given. To obtain systematic knowledge of the cavitation characteristics of the B-screw series in a uniform flow, tests were carried out in the large NSMB cavitation tunnel with a part of the B- screw series. The influence of cavitation on propeller thrust and torque was measured. n addition, inception curves were established for tip vortex cavity, sheet cavitation at suction and pressure sides, and bubble cavitation at back at midchord. The results of all these investigations concern- Table 2 Y~ R a r br = Cr D= r R= AL./Ao = Z= tr = Dimensions of Four and Five-Bladed B-Screw Series c~ Z ar tr b~ D Az/Ao c~ D c~ , O. 586 O, 24 O t, O. 524 O. 156 O ,582 O. 351 O, 72 O distance between leading edge and centerline at radius r distance between leading edge and maximum thickness of blade profile at radius r chord length of blade profile at radius r screw diameter radius tip radius expanded blade area ratio number of blades maxinmm thickness o1' blade profile at radius r ing the Wageningen B-screw series are given and discussed in this paper. Geometry of B-Series Screws A systematic screw series is formed by a number of screw models of which only the pitch ratio JP/D is varied. All other characteristic screw dimensions, such as diameter D, number of blades Z, blade area ratio Ae/Ao, blade outline, shape of blade sections, blade thicknesses, and hub-diameter ratio d/d are the same. The results of tests The Wageningen B-Screw Series 271

4 / [ [dynamometer --7 RPM t---~(propetter thrust[ /a"d torque) J Fig. 3 Measuring equipment for performing open-water test with the four-bladed B-screw series with bladearea ratios of.4,.55,.7,.85, and 1. and the five-bladed series with blade area ratios of.45,.6,.75, and 1.5 are given in this paper. Figs. 1 and 2 show the general plans of the fourand five-bladed screw propellers respectively. The dimensions of these screws are given in Table 2. The diagrams show clearly that the B-series screws have relatively wide blade tips, circularback blade sections near the tip, and airfoil sections near the hub. The four-bladed screws have a decrease in pitch at the hub of 2 percent in order to adapt the screw better to the velocity distribution behind a ship. A large number of experiments has shown, however, that the difference in efficiency due to this decrease in pitch in comparison with a screw with constant pitch is insignificant. The five-bladed screws have a constant pitch. The hub-diameter ratio of the fourand five-bladed B-series screws was d/d =.167. Open-Water Tests Test Procedure The open-water tests with the B-series screws were carried out with the usual apparatus shown in Fig. 3. The immersion of the propeller shaft was equal to the screw diameter. Before the tests were carried out, the system friction and dummy hub torque and thrust were determined so that the measured propeller thrust and torque could be corrected accordingly. The usual routine of open-water tests was followed; the rpm of the screw was kept constant, and by varying the speed of advance the desired value of the advance coefficient J was obtained. Usually the rpm was chosen as high as possible to obtain a high Reynolds number. The rpm was chosen in accordance with the maximum speed of the towing carriage and the capacity of the dynamometer used for the thrust and torque measurements. Most of the open-water tests were made at 45 rpm. The Reynolds number for the B-series screws, based on the chord length of the screw blades at.75 R, may be written as ~ C D "~-~ O rnln txp~ "X~O min ther / <. (rad) Fig. 4 Relation between drag coefficient CD and angle of attack ~x of equivalent blade profile of B 4-7 screw with P/D = 1 where Reo.wR = c vs~x/lza2 + ("75~rnD)2 c.7~ = AF,._, D A Z v = kinematic viscosity of water The diameter of the B-series screw models was chosen to be.24 m. At a screw rpm of 45 and with the advance coefficient J varying from to 1.5, the Reynolds number varies between Reo.7~R = (2.1 to 2.5) 1.A~ 1 vaoz This variation in Reynolds number is small and therefore it is permissible to base the Reynolds number of the different screw series on a mean value of J. The mean value of J is chosen to be 1.. With a kinematic viscosity of the water equal to v m2/sec, the Reynolds number for the different screw series becomes: v 272 The Wageningen B-Screw Series

5 1..g 1 KG. 8 K T ~.7 Q6.5 - i.4 Q3 Q2.1 Ol 2 3 / , O L 1~ Fig. 5 Open-water test results of B 4-4 screw series 1KQ K T ~o ' v Fig. 6 Ot Q J Open-water test results of B 4-55 screw series 1 Reo.7~ = 2.1 ~. "-. A Z According to this definition, the B 4-7 series screws, for instance, were tested at a Reynolds number equal to Finally, it must be noted that the tests with the B-series screws were conducted over a period of more than thirty years. These tests were carried out in different basins of NSMB and dynamometers with different capacities were used. Therefore, the Reynolds nmnber at which the different The Wageningen B-Screw Series 273

6 O~ 1K: E K T?o OE 5 O~ Fig. 7 4 g /, 15 Open-water test results of B 4-7 screw series 1G screw series were tested varied considerably. This fact must be taken into account when the results of different screw series are compared with each other. Analysis of Test Results Usually the open-water test results of a series of screws were faired (parameters: advance coefficient J, and pitch ratio P/D) and plotted in the conventional way with the coefficients: T KT -- pn2d 4 KQ-- Q pn2d 5 J Kr 27r Ko as functions of the advance coefficient J = VA/nD. By interpolating in the KT-KQ-J diagram of a screw series, most problems which arise when de- signing or analyzing screw propellers can be solved. From a correlation between the available design diagrams of the B-screw series, it appears that small differences exist. This is partly caused by Reynolds number effects, the degree of turbulence in the towing tank, and so on. Before making a cross fairing between the different B-screw series to blade area ratio and eventually to number of blades, we want to correct the test results for these effects. The effect of the Reynolds number on the test results can be taken into account by using the method developed by Lerbs [7] from the characteristics of equivalent blade sections. This method has been followed also by Lindgren [8], Lindgren and Bj~trne [9], and Newton and Rader [1]. Assuming, according to Lerbs, that the profile of the blade sections at.75 R is equivalent for the blade, simple relations between the coefficients K~, KQ, and J from the propeller tests, the corresponding lift and drag coefficients CL and CD and 274 The Wageningen B-Screw Series

7 /! B/ i " KQO~.... KT ~ \ 7Fi q\y_/~, ~ _ o al d2 ~3 Fig O / Open-water test results of B 4-85 screw series the profile angle of attack a can be deduced. As an example, the drag coefficients CD of the B 4-7 series screw with pitch ratio P/D = 1. is given in Fig. 4 on a base of angle a. From these results the values of minimum drag coefficients C~.ml, of the equivalent blade sections can be determined. n addition, the drag coefficients C~,m~. of the equivalent blade sections can be determined theoretically according to Hoerner [11 ] with : where C,,,,,, = 2C~-[1 +2 /o.w~ co.7~a _] Cf = drag coefficient of a fiat plate in a turbulent flow to.v~r/co.v~,u = thickness ratio of the blade section at.75 R The drag coefficient Cv depends on the Reynolds number and can be determined according to the 1TTC line. The thickness ratio of the blade section of B-series screws at.75 R is equal to to.v~,~/ Co.7~ = Z As A Z Hence the open-water test results of a screw propeller can be corrected for Reynolds number effects by shifting the experimentally obtained drag curve in such a way that the CD,mln according to the test results coincides with the theoretically calculated drag coefficient CD.,,,.. This theoretical drag coefficient corresponds with the assumed value of the Reynolds number. The concept is shown in Fig. 4. Once the lift and corrected drag coefficients C~. and CD and profile angle of attack a are known, the corrected thrust and torque coefficients KT and KQ and the advance coefficient J can be deduced. All the open-water test results with the B-series screws are corrected in this way. The assumed Reynolds numbers and drag coefficients C~,mi= are given in Table 3. To show the scale effect, the open-water test results with the B 4-7 screw The Wageningen B-Screw Series 275

8 - B 4-1 m OKQ KT % i F ~ O? 2 3 Fig. 9 O L 5 6 O?.8 9 1, J Open-water test results of B 4-1 screw series 1. Q9 OKo Og KT '~ 1?7 J ~ J J ~ E i,2 i i Og 5 O4 O3 O ~ 15 16,3 Fig. 1 Open-water test results of B 5-45 screw series 276 The Wageningen B-Screw Series

9 og 8 OKQ KT '~O ~ 3 ~ O~ g ~ O Fig. Open-water test results of B 5-6 screw series ~.~_~ KQ O8 KT NO.'7 O6 5 /. 3 O2. 1 b S g 7.8 g 1 12 ~3.L J Fig. 12 Open-water test results of B 5-75 screw series The Wageningen B-Screw Series 277

10 l \Z ,~ ~..... l KQ KT /.6 OS X x'v ~" "~7~-- ~--~ -- r ~1/ "--1 "-q O, O& ,2 Fig. 13 Open-water test results of B 5-15 screw series Table 3 Assumed Reynolds Numbers and Drag Coefficients CD, nl~. of Different Screw Series An 1.75 R Z As Reo.v,n 2 C c6.v, R CD,,ni~ X X t"7 1. X / X ~ X X Jo X ~ X [ X Reo.v, R = CD, rain = 2 Cp[1 -F- 2 [.TaR/tO.TaR] to.v6n CO.7SR 5.72AE/A6" 1/Z. 16 ( Z)Z 2.73 AE/Ao series are given for different values of the Reynolds number (see Table 4). The fairing of the corrected open-water test results with the B-screw series was performed with the aid of a CDC 33 computer by means of a Table 4 Reynolds Numbers and Corresponding Drag Coefficients CD, m~. to Which Results of B 4-7 Screw Series Are Extrapolated A E 1.Ta R Z Ao Reo.Tr, R 2 Cf c.75 R CD. mln (.3 X 16, l 1. X 16, X X 1 G ,lO X 16.41, regression analysis, Up till now only the fourand five-bladed B-screw series were analyzed in this way. To begin with, the results of the different screw series (parameters: advance ratio J and pitch ratio P/D) were analyzed, The thrust and torque coefficients were expressed as polynomials of advance coefficient J and pitch ratio P/D: KT = Ao,o + Aoa J-t- Ao,2 d Ao,, js The Wageningen B-Screw Series

11 , ~'li 7" i- Z[~-Z~ f.[ 1K KT ~.5. t Ot 3 O r Fig. 14 O , l,r* Open-water test results of B 4-7 screw series extrapolated to Reo.75 R :=.3 X loe; (C.. min ) P P P. -]--Zll,6~- j6_[_ Ax,o~-- + A,~) J-]- A1,2~) J2 +. pe 2 A2,~2 -~-... -t- A2,6~? j6. /36,P B6,6D~ p6 if6 KQ= d~6 Bo,o +Bo,1 J+i3o,,., J~+..+Bo,6 j6+ Bl,O~ B,1]~ ff -[- Bo 1,-/~.J"-] P o B1.6~- J) ff6_[_ p2.b~,o~ +... With the aid of a regression analysis the significant terms of the polynomials and the values of the corresponding coefficients were determined. From this analysis it was :found that the results of both the four- and the five-bladed B-series screws could be approximaled very well by the following series : P Kr = Ao,o + AoaJ + Ao,2:2 + Ao,J 3 + A,,o~ P 3 p2 p~ -1- A,~) J -t- - 11,3b J -lr-a2,1d~j...[- Ao,o~ p6 + A~a~J The Wageningen B-Screw Series 279

12 1K{ K T ~o ~ f Fig. 15.~ Open-water test results of B 4-7 screw series extrapolated to Reo.7a n = 3. X 16; (CD,mln :.78) KQ = BO,O + BoaJ + Bo,~J 2 + Bo,aJ a + B,,l P ~ J Kr = p2 p2 p6 + B~,a ~ P J" + B2, ~-2 + B~,a ~ ja + B6,o D6-- p6 + Bs,~?J Most of the terms with high powers of J and P/D were insignificant. Secondly, the results of the screw series with the same number of blades but with different blade area ratios were correlated with each other. The thrust and torque coefficients of the four- as well as the five-blade propellers were expressed as polynomials of J, P/D, and A E/A 8. As far as J and P/D are concerned, the choice of the terms was based on the foregoing analysis. The following polynomials were used for the calculation: Ko = A E P 6 A E2.P 6 + Co,o,o DP~66 + G,6, -~- ~ + G,~,O--A o 2 D 6- p6 A E p6 + Co,o,, + c,,o,, : ditto A Ea P 6 + Ca,~,o-- Ao 3 D 6 + Ce,<l Au2P6 CaG1A~ ap6 28 The Wageningen B-Screw Series

13 ~. v a i...., Qg~ 1K: 8 K T "r~o 7 \4) / O ~1 Q2 3 /, S "~ O "l~ 1.S 16 Fig. 16 Open-water test results of B 4-7 screw series extrapolated to Re.~5 n = 1 ; (CD. mi~ =.62) Again with a regression analysis the significant terms of the polynomials and the values of the corresponding coefficients were determined. For the four- and five-bladed B-series screws these results are given in Tables 5 and 6 respectively. Presentation of Test Results The results of the open-water tests with the four- and five-bladed screw series were extrapolated and expressed in polynomials. n these polynomials, either KT or K o were the dependent variable with the advance coefficient J, the pitch ratio P/D, and the blade area ratio A E/Ao as the independent variables. The form of these series together with their coefficients were given in Tables 5 and 6. These results can be used directly for solving problems which arise when designing and analyzing screw propellers if a computer is available. Also, we will give the results of the analysis in graphical form. With the aid of a tape-controlled drawing machine the coefficients Kr, Ko, and no were drawn in the conventional way as a function of J. The diagrams of the B 4-4, B 4455, B 4-7, B 4,-85, and B 4-1,;crew series are given in Figs. 5 through 9. Figs. 1. through 13 show the diagrams of the B 5-45, B 5-6, B 5-75, and B 5-15 screw series. The results of the screw series were extrapolated to different Reynolds numbers. The CD.,,in values of the equivalent blade section corresponding with the assumed Reynolds numbers were given in Tat)le 4. The KT-KQ-J diagrams are given in Figs. 14, 7, 15, 16, and 17. By interpolating in the'. K -KQ-J diagram of a screw series most problems which arise when designing or analyzing screw propellers can be solved. For design purposes various types of more practical diagrams can be derived from the Kr-Ko-J diagram. The most widely encountered design problem is that where the speed of ;advance of the screw VA, the power to be absorbed by the screw P, and the number of revolutions n are given. The The Wageningen B-Screw Series 281

14 _\ " t2 ~ ~ i i i... [... i... t. [ " i - ]..... B i =4: KT ~ 7 6 L ~ --i!! 1 Y o~ oi 2 "t~ , Fig. 17 Open-water test results of B 4-7 screw series extrapolated to Reo.vsn = 1s; (CD, m~ =.43) diameter D is to be chosen such that the greatest efficiency can be obtained. The problem of the optimum diameter can be solved in an easy way by plotting 7o and J as a function of KQ1/2/J /2 :: npl/~/va 5/~ The Taylor variable Bp is related to this dimensionless variable by the equation where B~ = 33.7 KQ1/2/J 6/2 = NP'/"-/Va V~ N = number of revolutions per minute ff = power in hp VA = speed of advance in knots. n the usual diagram, the design coefficient By is the base and a new speed ratio 6 is used. This speed ratio is defined as 6 -- ND v~ y in which D = screw diameter in feet The Bp-6 diagrams of all the four- and five-bladed B-screw series are given in Figs. 18 through 26. As a check of the correlation between the different diagrams, the efficiency 7o, speed ratio 6, and pitch ratio.p/d corresponding to optimum diameter are given in Figs. 27, 2S, and 29 on the base of B~. The curves of the four-bladed B- screw series are given in Fig. 27. Figs. 28 and 29 show the results of the five-bladed B-screw series and the B 4-7 screw series extrapolated to different Reynolds numbers. n cases where, Va, T, and n or VA, ]', and D are given, the problem of determining the optimum diameter or the optimum number of revolutions can be solved by plotting 7o and J as functions of KT/J 4 and KT/J The Wageningen B-Screw Series

15 Table 5 Form of Polynomial and Coefficients of Four-Bladed B-Screw Series Cx,y,z x :y g ~ ~ ~ # # ~ ~ ~ i # --i ~ ~ --i ~ ~ ~ ~ ~ ~ --i Dx,y,z " ~ ;5 ~ ~ ~ ~ ~ ~ ~ ~ ~ --2 KT = ZC~,>:[A~'/A,,]~[P/D]'[J] ~ KQ = ZD... [Au/Ao]~[P/DY[J] -" xyz i O Table 6 Cx,y,z Form of Polynomial and Coefficients of Five-Bladed B-Screw Series X 3' Z ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ T)x,y,z / KT, = EC... [AE/Aolx[P/D]~[J] KQ ~D:,u,~[AE/Ao].~[P/D]y[J] xyz For the different propeller groups the curves for optimum diameter (on base of ~ /KT/J 4) and optimum rpm (on base of v/kr/.l 2) are given in Figs. 3, 31, and 32 and Figs. 33, 34, and 35 respectively. Fig. 18 ~_ Open-water test results of B 4-4 screw series The Wageningen B-Screw Series 283

16 \A \ix x x _...,. Fig. 19 Open-water test results of B 4-55 screw series Fig. 2 Open-water test results of B 4-7 screw series 984 The Wageningen B-Screw Series

17 Fig. 21 Open-water test results of B 4-85 screw series Fig. 22 Open-wat,er test results of B 4-1 screw series The Wageningen B-Screw Series 285

18 o i o D to o o o h x'~ \4 g o # f Fig Open-water test results of B 5-45 screw series Fig. 24 The Wageningen B-Screw Series Open-water test results of B 5-6 screw series

19 O Fig. 2 5 Open-water test results of B 5-75 screw series Fig. 26 Open-water test results of B 5-15 screw series The Wageningen B-Screw Series 287

20 ?o O 5 5 O, ; 4q O: 3O 2O 3 &,- 2 JO 1.( P/c 3 D( Fig p Curves for optimum diameter of four-bladed B- screw series Bp Fig. 29 Curves for optimum diameter of B 4-7 screw series extrapolated to different Reynolds numbers 5O 3OO 8 -~ J Fig p Curves for optimum diameter of five-bladed B-screw series Fig. 3O \/, wd4 Curves for optimum diameter of four-bladed B.screw series Four-Quadrant Measurements Test Procedure To obtain data for ship maneuvering studies, it is necessary to know the propeller thrust and torque characteristics over the entire region of propeller operations. Data covering the entire operating region are scarce and most of the available data are related to isolated applications only. (See, for instance, [11], [12], [13], and [14].) Therefore, part of the B-series was tested over the entire range of operating conditions. Conse- 288 The Wageningen B-Screw Series

21 o~[ 7 5 ~o 5., T ',... B 4-4 ) i L -SO ! % 8 i C S 1 ~ ~ ~_K~T1 - "?- i i Fig. 31 Curves for optimum diameter of five-bladed B-screw series Fig. 33,3 Curves for optimum rpm of four-bladed B- screw series ~ 8 '% O5 i i /C $ % O~!, 2 3 ~ 5, , 2 Oe i, ~ Fig. 32 Curves for optimum diameter of B 4-7 screw series extrapolated to different Reynolds numbers Fig. 34 Curves for optimum rpm of five-bladed B-screw series quently, tile screw propellers were tested at the following combinations of speed and rpm: First quadrant: speed ahead, rpm ahead. The hydrodynamic pitch angle 3 (tan 3 = Vx/O.TrrnD) varies between and 9 deg. o ~ 3 4 9o Second quadrant: speed ahead, rpm astern o ~ ~ ~ lso The Wageningen B-Screw Series 289

22 ~ : To ',. 1 ~ 2 X as \ K/-~- T t 2 ] a Fig. 35 Curves for optimum rpm of B 4-7 screw series extrapolated to different Reynolds numbers Third quadrant: speed astern, rpm astern Fourth quadrant: 18 ~ ~ ~ 27 speed astern, rpm ahead 27 4 ~ ~< 36 The measurements were performed with the usual apparatus for open-water tests. The tests with the negative propeller advance velocities were conducted with reversely mounted model propellers and without change in the carriage motion direction. Within the normal test range of a screw propeller it is generally possible to perform the tests at a constant rpm. Outside this range, the tests must be performed at different rpm and speed of advance to operate within the range of capabilities of the dynamometer and on account of the maximum speed of the towing carriage. Usually the rpm and speed were chosen as high as possible to obtain a high Reynolds number. Presentation of Test Results The influence of pitch ratio P/D, blade area ratio AJAo, and number of blades Z on the characteristics of the B-series screws in the four quadrants have been determined. Therefore, open-water tests were performed with the following propeller models: 1 B 4-7 series screw with P/D = 1. 2 B 4-7 series screws with P/D =.5,.6,.8, 1., 1.2, 1.4 (influence of pitch ratio) 3 B 4-4, B 4-55, B 4-7, B 4-85, and B 4-1 series screws, all with P/D = 1. (influence of blade-area ratio) 4 B 3-65, B 4-7, B 5-75, B 6-8, B 7-85 series screws, all with P/D = 1. (influence of number of blades) The test results are given in Figs. 36, 37, and 38. n these diagrams the nondimensional thrust and torque coefficients: CT* Co* --- T ~r D2 ½o[V, fl + (.7~nD) 2] ~- ~p[va (.7~nD) 2] ~.D2"D 4 are given as function of the hydrodynamic pitch angle v~ fl = arctan-.7rrnd n Fig. 35 the influence of the pitch ratio on the characteristics in the four quadrants is given. Figs. 36 and 37 show the influence of blade area ratio and number of blades respectively. n ship maneuvering studies, it is necessary to have a mathematical representation of the data suitable for use on a computer. Therefore, calculations were carried out to represent the results of the four-quadrant measurements by a Fourier series. The thrust and torque coefficients Ce* and CO* were approximated by the following series: Cr* = ~ [A(K) cosflk+b(k)sinsk] K= CO* = ditto n Figs. 39, 4, and 41 the measured results and the approximation of these results by a Fourier series of 2, 5, 1, and 2 terms are given for the B 4-7 series screws with pitch ratios of.6, 1., and 1.4 respectively. From the results given in these diagrams it can be seen that the characteristics of the model propellers in the four quadrants can be adequately represented by a Fourier series of 2 terms. Tile values of the Fourier coefficients of the different screw propellers are given in Tables 7, 8, and The Wageningen B-Screw Series

23 c; -ll)cq 8 //:,%,,,.',, 4 /t "-..? -~. / ~8-12 \ \/ / \--..,// / / / ", \ /,,',,,,' -2C J -2~ / ~---~~ ; Fig. 36 Open-water test results with B 4-7 screw series in four quadrants J, 1, i.,j Cavitation Test Procedure Test Results To obtain systematic knowledge on the cavitation characteristics of the B-series screws in a uniform flow, tests have been carried out in the large NSMB cavitation tunnel with the B 4-$5, B 4-1, B 5-75, and B 5-15 screw series. The NSMB cavitation tunnel has a 9 cm X 9 cm closed working section and a uniform flow. The material used for the propeller models was normal propeller bronze. The surfaces were polished to a high-quality smooth finish. For reliable cavitation testing of propellers, the leading edges of the propeller blades must be very accurately finisbed to the correct shape. Therefore, all screw models were carefully corrected with a blade-edge microscope. A detailed description of this apparatus is given in [15]. The tests with the screw models were carried out at a constant water velocity in the test section of 5.5 m/see. The required range of cavitation numbers was obtained by wtrying the pressure in the tunnel. By varying the rpm of the model propellers a range of adwmee coefficients was covered between, and about 5 percent slip. Due to the normal restrictions of a cavitation tunnel it was not possiblle to cover the slip range up to 1 percent (zero ]). The air contenl: of the tunnel was kept within certain limits throughout the complete testing and ranged between about a/as =.2 and a/a~ =.33, according to Van Slijke. The cavitation inception curves were established by determining, as a function of the propeller load and cavitation number, the point at which cavitation just disappeared (desinent cavitation). This procedure was followed because the The Wageningen B-Screw Series 291

24 K CT* [ CQ* ~ Table 7 Coefficients of Fourier Series by Which Results of Four-,P/D.=.6 -P/D =.8..P/D = A (K) B(K) A (K) B(K) A (K) ~ ~ ~ / / ~ ~ ~ , ~ --' ~ ~ --i ;~ --i ~ i i i +.62 ~ ~ ~ i ~ --i ~ --i ;~ ~ ~ ~ ~ ~ i i ~ ~ i ~ ~ --i ~ ~ ~ ~ i # ~ ~ , ;~ ~ ~ # ~ ~ _o ~ ~ ~ ;~ ;~ ~ ' ;~ ~ , ~ , ~ ~" --t +, ~ ~ ~ +, ~ , t i ~ i i ~ ~ ~ ~ i ~ ~ -- +, i ~ i i i ~ i ~ # _o ~ ~ ~ ,26454 ~ ~ ~ ~ --i ~ ~ ~ --i ~ --i # ~ i i # ~ ,58492 ~ ~ ~ , i # ~ ,68631 ~ ,1163 ~ ~ ~ # ;~ --i ~ ~ ~ ~ i ~ ~ ~ ~ ~ --i --.57,544 ~ , , ~ , determination of the point of incipient cavitation does not give cousistent test results whereas desinent cavitation is fairly repeatable. Analysis of Test Results The net propeller thrusts and torques measured in the tunnel were converted to the conventional nondimensional thrust and torque coefficients KT and KQ. These coefficients were calculated as a function of the advance coefficient J = VffnD and the cavitation number - -- p~ - pv The effieiencies were calculated from the relation ~ ~ J 27r Kr KQ using the KT and Ko values from the curves faired through the measured values. The cavitation inception curves for the tip vortex cavity, the sheet cavitation on pressure and suction sides, and the midchord bubble cavitation were given on a base of the physically realistic thrust coefficient ~P V~2 1" 8 Kr ~r D 2 7r.]2 and the cavitation number ao. Corrections were applied to the water velocity and the pressure in the working section for tunnel wall interference. These corrections were based on the identity of the thrust coefficient KT for the screw models in the tunnel and that from the open-water tests. The applied corrections are in principle the same as indicated by Burrill [t6]. The correction leads initially to the relation Jt = kj. The factor k is determined for every screw as a function of J for equal KT, from the open-water KT-J 292 The Wageningen B-Screw Series

25 Quadrant Measurements with B 4-7 Screw Series Are Approximated 1.,.P/D = P,/D = 1.4 B(K) A(K) B(K) A(K) B(K) +. ~ t ~ ~ t ~ ~ t ~ ~ ~ ~ ~ ~'~ ~ ~ cff # ~ ~ ~ ~ ~ ~ ~ # # # ~ ~'~ # ~ # # ~ # # # # ~'~ # # # # ~ ~ ~ ~ ~ ~'~ (,)7 ~ # # # ~ ~ ~ ~ ~ --i ~'~ # # # # ~# --' ~ # # # ~'~ ~ ~ ~ ~ ~'~ --' ~ ~ ~ ~ ~ ~ --i ~ ~ ~ ~# ~ ~ :1 ~ --' ~ ~ ~ ~ ~ _o ~ --' ~'~ ~ ~ ~ ~ ~'~ ~ ~ ~ --' ~'~ # _o ~ ~ ~ ~'~ to ~ # to ~ ~ ~ ~ ~ ~ ~ ~'~ ~ ~ ~ --' ~ ~ ~ --1 ") +._1_3 ~ +.295(; ~'~ ~ ~ ~ ~ ~'~ ~ ~ ~ ~'~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~'~ ~ ~ ~ ~ ~a # # ~ ~ ~'~ ~ :) ~ ~ ~'~ --' # _o ~ _o ~ ~'~ # ~ --' ~ ~ ~ ~ --' ~ --' ~ --o ~ ~ ~ ~ q7 ~ ~ ~ (,)9 # ~ --' # # ~'~ ~ ~ ~ ~ ~'~ --' # # ~ ~ ~ ~ ~ ~ -- ") ~ ~-* ~ ~ ~ ~ --2 relationship and non-cavitation influenced Kr,- oft relationship measured in the cavitation tunnel. (The subscript t refers to the cavitation tunrtel experiments.) From the foregoing comparison of open-water and cavitation tunnel test results the relationship between KT,,/Jt 2 and KT/J 2 was determined for constant values of the correction factor k. This relationship was used as a base for the corrections of J and ao in the regions where cavitation influenced the thrust delivered by the propeller. An extra, correction had to be applied to the Kot-J, relationship for identification with the open-water KQ-J relationship. For the correction of the cavitation-influenced Kot curves it was assumed that the ratio between the influenced and not-influenced torque coefficients for a given cavitation number remained the same, after application of the total correction. Because of the relatively low ratio of propeller disk area/area of the working :section of the tunnel (about.71), the corrections on velocity and pressure for tunnel wall interferec, Lee were small. Presentation of Test Results The influence of cavitation on propeller performance of the B 4-85, B 4-1(}, B 5-75, and B 5-15 screw series i~ given in ]Figs. 42, 43, 44, and 45 respectively. [n these diagrams KT, Ko and 7o are shown as functions of J with the cavitation nulnber ao as parameter. The faired eurw:s for the onset of tip vortex cavitation, suction- and pressure-side sheet cavitation, and midchord bubble cavitation of these series are given as a functiou of the cavitation number o and the thrust coefficient Cr in Figs. 46 through presenting these results the following abbreviations for the different types of cavitation were used: tvc = tip vortex cavitation The Wageningen B-Screw Series 293

26 Cr* Table 8 K [ tlo Coefficients of Fourier Series by Which Results Are Approximated of Four-Quadrant A~'/Ao =.4 AE/Ao =.55 ~.4E/Ao = A (K) B(K) A (K) B(K) A (K) -} # --1 -}-. # ~ --1 -}-. # -}- -} ~ ~ ~ ~ -}-} ~ ~ --2 -} ~ --1 -} ' # --1 -} ~ --1 -}-} ~ --1 -}-} ~ ~ ~ '2 -} } ~ --3 -} ~ ~ --2 -} ~ --1 -} ~ ~ ~ --2 -} ~ ~ ~ --2 -} ~ # ~ --1 -} ~ # ~ --2 -} ~ --2 -} ~ ~ ~ --2 -} ~ ~ ~ ~ --2 -} ~ --2 -} ~ ~ --2 -} ~ ~ _o -} ~ --3 -} } ~ ~ _o ~ ~ ~ ~ --' ~ ~ ~ ~ ~ --2 -} ~ --2 -} # --2 -} ~ ~ --2 -} ~ ~ ~ ~ ~ ~ ~ --4 -} ~ --3 -} ~ ~ --2 -} ~ --1 -}-. ~ ~ --1 -}-. ~ + -} # -} ~ ~ ~ ~ ~ --1 -} ~ ~ --1 -} } ~ ~ ~ --2 -} ~ ~ ~ # ~ --1 -} ~ -} ~ ~ :~ --2 -} ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ --1 -} ~ --1 -} ~ ~ ~ --2 -} ~ --1 -} ~ }-.136 ~ } , } ~ --' }-.184 ~ --1 -} ~ ~ --2 -} ~ ~ ~ ~ ~ } ~ --2 -} ~ --1.}..73,531 ~ ~ --2 -} ~ ~ --2 -} ~ ~ ~ ~ } ~ --3 -} ~ --2 -} ~ ~ ~ --2 SEE ssc = sheet cavitation on suction side psc = sheet cavitation on pressure side bmc = bubble cavitation at midchord of suction side. Further to the overall presentation in Figs. 46 through 49, Figs. 5 through 53 show the complete results of the cavitation inception measurements for all propellers with ])/D = 1.. The suction-side cavitation developed from the tip vortex cavity and spread down over the blade when the loading of the screw was increased. The following indications were used to describe the extent of the cavitation: ssc 1. R: beginning of suction-side cavitation at the blade tip ssc.9 R: suction-side sheet cavitation spreading down from blade tip to.9r ssc.7 R: suction-side sheet cavitation spreading down from blade tip to.7r. The pressure-side cavitation started at the leading edge between.6 _R and.7 R and extended along the leading edge to both sides when the loading of the screw was decreased. Conclusion 1 The derived polynomials of the thrust and torque coefficients of both the four- and fivebladed B-series screws and the Fourier analysis of the "four-quadrant" measurements enable design calculations and analyses with a computer. 2 The cavitation inception diagrams and the diagrams showing the influence of cavitation on thrust and torque can be used for making quasisteady predictions of the cavitation characteristics of a screw in a nonuniform wake flow. 294 The Wageningen B-Screw Series

27 Measurements with B 4-4, B 4-55, B 4-7, B 4-85, and B 4-1 Series Screws with P/D = 1..7~ B(K) "AE/A(, = A (K).85- B(K) A (g) "Az/Ao = 1. B(K) # # # # ~ # ~ , ~ ~ --1 +,1364 ~ # ,15776 ~ ~ # ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ s ~ -1 TABLE ~ _o ~ ~ ~ ~ ~ ~ ~ # ~ C ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ # ~ ~ ~ ~ --') # ~ ~ ~ ~ ~ ~ ~ c,5 ~ ~ ~ ~ '5 ~ ~ ~ --i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i ~ --i --.2'222 --i i ~ ~ ~ ~ ~ ~ ;~,2 ~ ~ ~ ~ ~ ~ ~ (17 ~ --' ~ ~ ~ ~ ~ ~3 ~ ~ ~ ~ --') --,94578 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (15 ~ ~ (; ~ l ~ ~ ~ The cross-correlation between the design diagrams, as has been done in this paper for different blade-area ratios at constant blade number, can be extended to different blade numbers. n addition, the effect of the Reynolds number can be taken into account in the polynomials for Kr and KQ. 4 Hence, the effect of the Reynolds number on the screw characteristics being known, the consequences of this effect can be introduced into trial predictions. 5 Further progress on these subjects to be made at the NSMB will be discussed in a second paper. Acknowledgments The authors are indebted to many members of the technical staff of NSMB for their contribution to the contents of this paper. n particular, they express their appreciation t.o Dr. H. le Grand for performing the numerical calculations. Special mention should be made of the contribution of Mr. H. Nijding, who did over a couple of years a large part of the work of calculating and preparing the diagrams. References 1 G. S. Baker and A. W. Riddle, "Screw Propellers of Varying Blade Sections in Open Water," Trans. NA, 1932 and G. S. Baker, "The Design of Screw Propellers with Special Reference to the Single-Screw Ships," Trans. JVA, 1!-)34. 3 L. Troost, "Open-Water Tests with Modern Propeller Forms," Frans. NEC, 1938, 194, and W. P. A. van Lammeren, L. Troost, and The Wageningen B-Screw Series 295

28 Cr* G, Table 9 K o L Coefficients of Fourier Series by Which Results Are Approximated of Four-Quadrant A(K) O ~ ~ ~ ~ ~ ~ ~ ~ ~ -i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ --i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ B 3-65 B(K) ~ B A (K) 4-7--~ B(K) -B 5-75 A (K) # ~ ~ ~ ~ # o ~ # ~ ~ # # SEE TABLE ~ ~ ~ ~ ~ ~ # ~ ~ ~ ~ # # ~ ~ ~ ~ # # # # # ~ ~ # ~ # ~ ~ ~ ~ ~ ~ # ~ ~ # # ~ ~ --i ~ ~ ~ --i ~ ~ ~ ~ # o1 # ~ ~ ~ _o ~ ~ ~ ~ ~ ~ ~ ~ # # ~ ~ ~ --2 J. G. Koning, Resistance, Propulsion and Steering of Ships, H. Stare Haarlem, Netherlands, PrinciplesofNavalArchitecture, SNAME, New York, J. D. van Manen, "Fundamentals of Ship Resistance and Propulsion, Part B," Publication No. 132a, NSMB. 7 H.W. Lerbs, "On the Effects of Scale and Roughness on Free Running Propellers," Journ. ASNE, H. Lindgren, "Model Tests with a Family of Three and Five-Bladed Propellers," Publication No. 47, SSPA, H. Lindgren and E. Bj~trne, "The SSPA Standard Propeller Family Open-Water Characteristics," Publication No. 6, SSPA, R.N. Newton and H. P. Rader, "Performance Data of Propellers for High-speed Craft," Trans. R[NA, H.F. Nordstrom, "Screw Propeller Characteristics," Publication No. 9, SSPA, Ya. Miniovich, "nvestigation of Hydrodynamic Characteristics of Screw Propellers Under Conditions of Reversing and Calculation Methods for Backing of Ships," BUSHPS Translation 697, K.. Meyne, "Umsteuereigenschaften yon Sehiffspropellern," Schiff und Hafen, Heft 5, Sv. Aa. Harvald, "Wake and Thrust Deduction at Extreme Propeller Loadings,"Publication No. 61, SSPA, J. H. Witte and J. Esveldt, "The Blade Edge Microscope of the Netherlands Ship Model Basin," SP, Vol. 13, L.C. Burrill, "Tunnel Wall nterference," Appendix 7, Seventh nternational Conference on Ship Hydrodynamics, Publication No. 34, SSPA, The Wageningen B-Screw Series

29 Measurements with B 3-65, B 4-7, B 5-75, B 6-8, and B 7-85 Series Screws with P/D = 1. B 6-8 E; 7-85 B(K).4 (K) B(K) A (K) +. ~ # # ~ # # # +.1&~99 # # # # --'2 +.24!)84 # --i ~ ~ ~ --i ~ ~ # ~ --i # --i ~ ~ ~ ~ ~ ~ ~ --' ~ --'; ~ ~ ~ :_)37 ~ ~ ~ --' ~ --' ~ --' ~ ~ ~ ~ --' ~ --' ~ ~ ~ ~ ~ ~ ~ ~ ~ t74 ~ ~ ~ ~ ,585 ~ ~ ~ --' ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ --' # ~ # '931 ~ --' ~ ~ --2 % ~ ,9162 ~ --' ~ ~ ~ _o ~ _o ~ ~ ~ # ~ + +. ~ + +. ~ + +. ~ +O ~ ~ ~ ~ ~ ~ ~ ~ # ~ ~ ~ O ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ --i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _o ~ ~ ~ ~ ~ ~ ~ --' ~ --') # # ~ ~ _o ~ ~ ~.~ ~ _o # # # # # # ~ ,q22 # # # -' # ~ ~ # # # ~ # # --2 B(K) ~ ~ ~ ~ ~ ~ ~ -t # ~ ~ ~ t ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Eric Bj/:irne, 4 Visitor: First of all must express my satisfaction over the mathematical fairing of the test results. The divergence between different issues of the Wageningen propeller charts has been a source of long and often hot discussions about propeller dimensions and eflicieneies. The regression and Fourier analysis of the test results are of great value for those working with analogue and digital computers. However miss open-water test results presented on the basis of KT/J ~, which sometimes can be very useful at the project work. Speaking of parameters it seems inconsequent that the nondimensionless factor/3p is still used while the factor B~ is now replaced by V/~Tr/Jfl therefore 4 The Swedish State Shipbuilding Experimental Tank, G6teborg. Sweden. Discussion suggest that a further step in the development will be replacing B v with %/-Ko/.l ~. Also, the &values appear in some diagrams instead of the advance ratio J. The influence of Reynolds number, R,,, on the optimum propeller diameter, pitch, and efficiency is interesting to notice, but the difference between model and ship propellers with regard to drag coefficients must not be overestimated because of partly laminar flow in the propeller model region of R,,, Fig. 5'4. The gas conten L of water, {n the cavitation tests, ct/~t.~ =.2-.33, is somewhat higher than what is usual in the SSPA tunnel. Differences in gas content nmy influence the inception of cavitation, especially on the tip vortex cavitation. (text continued on page 312) The Wageningen B-Screw Series 297

30 C[ 1C P/D =1 B4-B B4-55 -B4-4 -'4 :E Q o B4-7 ~ z.o /,//J... /" i q o \,,ii,j,,,",,,,,;/ ~ ~-~-~-;'~,~:di~i : '~!, / /3 2 ~ Fig. 37 Open-water test results with B 4-4, B 4-55, B 4-7, B 4-85, and B- 4-1 series screws with P/D = 1. Fig. 38 Open-water test results with B 3-65, B 4-7, B 5-75, B 6-8, and B series screws with P/D = 1.

31 FOURER SERES OF 2 TER~4S _,! ] / / o / ~ 866 Fig. 39 Result of four-quadrant measurements of B 4-7 screw with P/D =.6 approximated by Fourier series of 2 terms CURVES OF C T AND CQ APPROXMATED BY A FOURER SERES OF 2 TERMS cl ~ ~---~-'~ -16o ,, g 32 36" Fig. 4 Result of four-quadrant measurements of B 4-7 screw with P/D = 1.O approximated by Fourier series of 2, 5, 1, or 2 terms The Wageningen B-Screw Series 299

32 ! t i FOURER SERES OF 5 TERMS ~2 o8 c~ -OC~ _//_... _~_ 4! Jc;,# /3 2i Fig. 4 (continued) 2 i FOURER SERES OF 1 TERMS _1 / ii i c~_! _ ! J, [ 4 BO /3 2o 2& e Fig. 4 (continued) 3 The Wageningen B-Screw Series

33 FOURER SERES OF 2 TERMS 1E....,!! j 4Q 8 12e 16 / Fig. 4 (continued) L i FOURER SERES OF 2 TERMS c~ -1c~ oz -4 / / "\ \ / :~ 2 2~!,\! i / /_ / /,Y f, i '3: Fig. 41 Result of four-quadrant measurements of B 4-7 screw with P/D = 1.4 approximated by Fourier series of 2 terms The Wageningen B-Screw Series 31

34 ~ K T 1OK O ~p ~5 B z,-85 ~.oe l ' i (~.3 f,.5 ~.7 g g 15 Fig. 42 nfluence of cavitation on thrust and torque of screws of B 4-85 series B z,-85 :,o 8 7 i f~ 3 ~".3 /, QB ~g J A 1.5 Fig. 42 (continued) 32 The Wageningen B-Screw Series

35 Q9 B ~-B5 %:L2 \ K T OKQ % ~f 2 ~ i---~ X O g.6 (17 3 ~ J Fig. 42 (continued) 1.1.~ " \ \ B 4-85 P/D =1.4 t i f \ K T OKQ % ~o ~ 2 O~ 5 6 CLB 9 1. J Fig. 42 (continued) A The Wageningen B-Screw Series 33

36 - i~ii.~\\~ E o~ 5 - Oe ~D =O8 OKQ "gp a~..co o, ~ -.'b.. o ~ t13,4 5 Fig Q t2 1,3 t,4 J nfluence of cavitation on thrust and torque of screws of B 4-1 series 1.1 B 4-1 P/D" 1..9 E KT 1K O "Tp o.~ ~P ~-o=~ ~ o~ 3.& Q8 9 1,,3 Fig 43 (continued) ks 34 The Wageningen B-Screw Series

37 B a-oo %=,2 K T 1 KQ 'gp 4 5 Q ~ tl 2 13 J 1~5 Fig. 43 (continued) K T 1OK Qi 9p 5- %.5 6 O7 8 Fig. 43 Q9 ~,g 12 t.3 t~ 1.5 (continued) The Wageningen B-Screw Series 35

38 B 5-75 P/Do o 8 "~p O-o >> \ 6 K T OKQ % 5 ~ o.1 "-"~"-'- --'-~"2" -~.~.~. / -'~._~-~-f~ ~ ""- "" 7"--. / t ~, Fig. 44 nfluence of cavitation on thrust and torque of screws of B 5-75 series B 5-75 P/'D = to K! OKQ % _q J Fig. 44 (continued) ta t5 3 6 The Wageningen B-Screw Series

39 OSf T" - ~ T B 5~75 l< T O KQ 5 Oz O~! ol O O t3 1 ~, 1,5 Fig 44 (continued) B 5-75 P/D 14 ['(T ~o:4 o ")p i j ' p j 3 O.4 5 Q6 7 ( t 12 J Fig. 44 (continued) [3 it. S The Wageningen B-Screw Series 37

40 K T 1K o % ~:: i B 5-15 P/o:o~ i r OB 9 [ Fig. 45 nfluence of cavitation on thrust and torque of screws of B 5-15 series B 5-15 %:,o K T t% % E /, -~ Ck3 y' ~f O!, O.S 6 ( to 1.1 J J 12 L3 if, 1,5 Fig. 45 (continued) 38 The Wageningen B-Screw Series

41 B 5-15 ~3 =12 K T OKQ '~ po-o.>> Co.L2a ~ ~, / / / / / /./ / \ / / / ol Q3 4 5 ~f, i r 6 Q7 Q8 9 1 g 11 1,2,\ ,5 Fig. 45 (continued).... [ B S-lOS P/D =1/~ ~T 1< OE ~?P OE ] "'\~ \ Q3.5-6 q7 [18 og 1. 1] J Fig. 45 (continued) L4 1S The Wageningen B-Screw Series 39

42 ! ~ J ---l-- B,-Z r-1 ~/ / // $ F o~kl i -T1 1- i--i ]- ;/i i 1-]i 5 7 OJ Q z; CT l~ig. 46 Cavitation phenomena on B 4-85 series screws [ O5,7 Fig. 48! CT Cavitation phenomena on B 5-75 series screws ps ~P/D "8 -. ~_\o ~, ssc OR t-/,,1l "~ i! / //.... %#.8-1 //. i, _!,_~L_ P/F=O8-15 o ~%-11 o.~ ~,o-o,?'j'~/,~oj 1[, " i /,! - \~ ~ ",, //X--ll i i~.7 7~ bmc at{ p tch r 4 tlo$ 5 -- Oz. -- o, 1 Q: 5 ~7 O1.2 CT 3 OZ o:!-,... t- o, _~ Oz. 5 Q CT Fig. 47 Cavitation phenomena on B 4-1OO series screws Fig. 49 Cavitation phenomena on B 5-15 series screws 31 The Wageningen B-Screw Series

43 ~ ! 2O i Bs?5 P~=! 71---, \ 3F -- [ + 2! 7' t ('Q5 ~ Of '-. 5 CT , C T Fig. 5 Cavitation phenomena on B 4-85 series screw with P/D = 1. Fig. 52 Cavitation phenomena on B 5-75 series screw with P/D = l.o r / 4c ' L B 5-1S psc 7 -~ a'o ~< _Z o~ ~ ~ 3mc, air PlltCh rahos o+: --i--[ o.~ o, Q5 ] OZ CT Fig. 51 Cavitation phenomena on B 4-1 series screw with P/D = 1. Oos 7.1 (~2 3 ~. 5 7 CT / ~ _21~_ Fig. 53 Cavitation phenomena on B 5-15 series screw with P/D = 1. The Wageningen B-Screw Series 31 i

44 O.Ot o "%. water test with ~eomot~: s~milor ~/er-mode/s log R n 75 Fig. 54 Minimum drag coefficient The cavitation Emit curves given by the authors were determined on the basis of desinent cavitation instead of incipient, due to the better repeatability for the former type, which means varying pressure at a constant rate of revolutions. Another method to determine these boundary curves is to keep the pressure (i.e., cavitation number, ~) constant and vary the rate of revolutions (advance ratio). The results of incipient cavitation tests to some extent also may be faired by the aid of Lerbs' method of equivalent profile. n Fig. 55 local critical cavitation numbers, ~.7~ = ~o/( /J2), representing results from tests with the SSPA standard propeller models 5.6 (see the authors' reference [9]), with different pitch ratios are plotted versus the lift coefficient, CL, for the profile in question. Faired mean curves through the points will indicate the cavitation limit and may be transformed to ~-J curves by knowledge of the CL-J Correlation. The test results have been compared with calculated values of the local dimensionless pressure difference, Ap/q, at certain points on the equivalent profile (2-dimensional flow). The correlation, CL ap, gives a guidq ance for the above-mentioned fairing. S. Curtis Powell, Member: will congratulate the authors by observing that they have contributed a quantum step in our knowledge of propeller characteristics, particularly in four quadrants and under cavitating conditions. object to the authors' statement about the most widely encountered design problem on three counts : (a) in this country the revolutions usually are not given, in contrast to Europe where internal-combustion engines are more common; (b) the diameter should be optimized, which is not the same thing as maximizing the propeller efficiency; and (c) the speed is not given, but is that speed at which the ship will operate with the given power; although the diameter is not likely to be seriously mischosen if a reasonable estimate of speed in service is made. (This is not the specification speed and power.) Although the diameter may not be significantly in error, other () 8 o oo~ Local pressure nc. cavit. _ o ~j~, 2-dim. flow test results "//~ ~/f/.\ - - y/c = 5 ~ Back cavt totion r sheet o o,'//~ "//~ O ". "... " ~ bubble ~3 ~; o.6 ~,,,\'/'....5 o Face -"- _ ///\ o ~#lc.48, o..~,,~,,,,\,,~\\\><\\,-\\\' () -J ~ O2.9.s J :... :.., re,'on \o Q Ltft coeffic/en t, C L 3.~ Fig. 55 ncipient cavitation at.75r, SSPA standard series 5.6, P/D = The Wageningen B-Screw Series

45 quantities involved in the ship-propeller-engine match may be more seriously affected. n any case, iteration is clearly in order (see Section 17, Chapter V, Principles of N~tw~.l Architectlt.re). much prefer to work with the Kt-J system of charts as the match achieved there is philosophically and practically more comfortable. The Bp charts in the present paper are not consistent with those of the aforementioned reference to PN.4, especially as regards the best efficiency line. Since both sets of charts understand to be derived from the same source, would appreciate it if the authors would give us an explanation for this discrepancy. The numerical constant relating Bp to K~ and or seems to imply operation in fresh water. This is an additional complication in the use of the B~,-a charts. would like confirmation that freshwater operation is indeed implied. To those familiar with the Russian results [12] for operation in four quadrants, the use of inverted Co scales in Figs. :39 to 41 lnay be confusing at first. Would the authors confirm that their conventions take thrust, 7", as positive when it is in the direction to propel the ship ahead and torque, Q, positive when it is in the direction corresponding to normal ahead propulsion? further assume that the 2-term Fourier series coefficients are obtained from a least-meansquare fit to a number of input points much in excess of 2. Can the authors tell us what is this nmnber of inputs? The use of "hydrodynamic piteh angle, fl," for presentation of four-quadrant data is a clever proeedure to eliminate some of the troubles with plots made to a base of J (and 1/or) or RPM. William B. Morgan and Geoffrey (3. Cox, Members: This paper presents very useful information for the naval architect who must make predictions of propeller performance. t represents the most extensive and comprehensive amount of series data known to us. The inclusion of a rational modification to the data to allow for the effects of Reynolds number on the propulsion characteristics is most welcome. Some of us have been concerned about the problem for a long time, especially for screw series data which are based on relatively small-sized models. We think that the cross-fairing technique used in the paper has greatly enhanced the Wageningen B-screw series data as there were obvious differences in some of the data presented in the past. We consider it a pity that the authors did not take the opportunity when preparing this paper to present Figs. 1S through 26 in nondimensional form. The use of these diagrams may require a correction for the density of the fluid medium in which the propeller will operate. For instance, are the Bp-~ diagrams presented in this paper for the propeller operating in fresh or salt water? t is very doubtful whether the cavitation data presented in Figs. 46 through 5:3 are useful or whether Conclusion 2 regarding cavitation inception is valid. Studies of the. many factors influencing correlation of cavitation inception made by the Cavitation Committee of the TTC would tend to invalidate any application of model propeller methodical series data to the prediction of cavitation inception on a full-scale propeller operating behind a ship. The authors have defined fl incorrectly in that it should be referred to as advance angle and not hydrodynamic pitch angle. The hydrodynamic pitch angle B~ includes induced effects. ncidently, why haven't the authors used standard TTC notation for all of their symbols? J. G. Hill, Member: t is always a privilege to see another propeller paper front Wageningen because of their high quality and completeness. This paper is no exception and provides a wealth of information to those concerned with propeller problems. t is noted that the B4-7 series was tested at a Reynolds nnmber of 3.5 X 15 and the assumed Reynolds Number for the B4-7 chart is 1. M 1/) 6. On the basis of Fig. 29 the efficiency for a given Bp should be higher for the higher Reynolds number of the new corrected presentation compared to the actual test results, as shown in Principles of N~twzl Architectl~re. Actually the new curves corrected to the higher Reynolds number show lower efficiency than the original curves. Figure 29 also indicates that delta should be higher and the pitch ratio lower with increasing Reynolds number, but the curves show the reverse. f these discrepancies are the result of the computer fairing, it would be interesting to know how much the test data was departed from and the justification for the shift. Since the linear ratio from ship to model is frequently about 3 for large ships, the Reynolds Number changes by a factor of about 15. Figure 29 indicates about a 1-percent increase in efficiency for this change in Reynolds number. Do the authors expect that this gain will be realized with the ship propeller or are there other factors that will prevent this gain? n Resistance, Propulsion and Steering of Ships it is noted that "this increase in efficiency is partly or wholly cancelled" ['.4.]. The cavitation data and maneuvering data are The Wageningen B-Screw Series 31 3

46 a valuable addition to the literature and it is hoped that the next paper will cover the cavitation of the lower blade areas and the 6- and 7-bladed series. The cavitation comparisons in Figs. g6 to 53 demonstrate that each combination of pitch and camber results in optimum performance at one particular thrust coefficient. Therefore, for any specific design condition the camber and pitch must be varied from the series values to obtain satisfactory performance. Particular attention must be paid to the inner radii of propellers for high-speed, twin-screw ships, where cavitation erosion is frequently found on the pressure face of constant-pitch propellers. P. C. Pien, Member: The results of the Wageningen B-screw Series given in this paper are somewhat different from. those previously published. This is a consequence of the refairing and the correction of Reynolds number effect. By comparing the model propeller test results in Fig. 7 and the extrapolated full-scale results in Figs. 16 or 17, we notice that the efficiency of a full-scale propeller can be as much as 2 percent, or more, higher than that of a model propeller. This is a delightful surprise. Perhaps before we accept these extrapolated results with confidence, we should examine their extrapolation method rather carefully. The extrapolation procedure used in the paper was developed by Lerbs almost twenty years ago. His basic idea is that the efficiency of any blade section can be considered as a product of the idea efficiency and the profile efficiency. The idea efficiency of a blade section is a function of the loading and the advance coefficient X. t is independent of the section profile geometry. The profile efficiency, on the other hand, depends upon the profile geometry as well as the loading. This is in line with the fact that propeller efficiency can be considered as a product of propeller idea efficiency and the propeller blade efficiency. t is possible to choose a blade section, the characteristics of which can be used to relate the openwater characteristics of the propeller. This can be done rather easily as shown in reference [7] of the paper. This chosen section is called the equivalent profile of the propeller. So far, there should not be any question. t is just another way of presenting an open-water test result. Then, two assmnptions are made in order to account for the Reynolds number effect as follows : The first assmnption is that the.75r section can be chosen as the equivalent profile with true physical meaning and that its characteristics vary with Reynolds number the same way as that of the propeller. The second assmnption is that the profile drag can be divided into two parts, the tangential component due to skin friction and the normal component due to pressure distribution and that only the first part has Reynolds number effect. The first assumption may be reasonable since the center of propeller loading is in the neighborhood of.75r and the profile geometry does not change rapidly there. However, the second assumption is quite contrary to the test results of the wing section. Figure 56 shows the drag coefficient of a NACA 66-6 wing section. The upper curve is for roughed section. The lower three curves are obtained at three different Reynolds numbers. The minimum drag coefficients of the three curves are the same while the shape of the curves are quite different. This means the minimum drag coefficient of a profile is independent of Reynolds number, and the pressure drag coefficient is highly dependent on Reynolds number. This situation seems to contradict the assumption made in the extrapolation procedure. Figure 57 shows the same tendency of a slightly cambered section. n view of this would be inclined to question the accuracy of the extrapolated results, especially for the heavily loaded conditions. J. B. Hadler, Member: would like to add my thanks, along with others, to the authors for their efforts in reanalyzing and expanding the test work on the 4- and 5-bladed propellers of the B- Screw Series. The removal of inconsistencies in the old test results along with the inclusion of the quadrature and cavitation measurements are welcome additions. The inclusion of the coefficients for the equations fitted to the test results will be of material assistance to those of us who use the digital computer in design or analysis work. My comments on this work will be restricted to questions of clarification or amplification. 1 Do the authors plan to extend this work to the 6-bladed series--an area of growing importance to the merchant ship propeller designer? 2 Do the authors plan to extend the derived polynomials for Kr and K o to include the effect of cavitation? 3 The paper indicated that the equations derived for KT and t o approximated the experi- mental results "very well." Could the authors give us an idea of the quality of the fit through the usual statistical terms such as standard deviation? 4 The authors' study of the effect of Reynolds number upon the open-water performance, particularly the influence upon the optimum diameter and propeller efficiency, is of sufficient magnitude to make the direct use of the design charts based 31 4 The Wageningen B-Screw Series

47 - - " " ~ :l.2 [,, 'i-? ".. o Q (Q R ~Q R r3o CO,Y o.z.4 -- m ~.ol ~.o~ Z Z o 'c/c i -t.~.8 i -'~! i i ~k - ~,,~ -.~ 5, t ~"-.O ~3)9 ~ - - ~t '5,) ""-.. Sfondal'd Standard, rouqhmexs --- OSOc s/i~uloted sph't flop def/acfed v ~ --. Src~do~d rouqhness - *'~ i.o2o /.O --- L--L-- i-- i i i' / / i, i ~, t ": ~.4.8 /.2 Section tiff coefl~c~f~ C~ ii -1 L8.Z i ~ k---- i -- i.o,~ - (a ,#.o/z - - ~.OOO.oo~ Z- Z O, ~ ~5 4.s -zi.4 ~/c.g,8 /.' T - i-! - ~! " -;4- i.. ~_ il i -- ~ 4 Z]Z- i, L~ Z- Z : - Z: --, t - i: "' J i ~ f.,, ' T po~ -.4-,1~. ~ -- x e 3"-- y/c. ~ o O.O,tlO.---_25' F 6. --J ~ 4--- L -- ~ -.--~-- o g.o_l..zsrl-.o/t, ; O,2e Mmu~/ed spilt floe de flecfe /1: v ~.... v', ~!n lo~ "d rouqhtless "~ i i " -L ~ i ' i;i ~ i Li ,~ d $e 't/o rift,:oefflclenf,, c fig. 56 Drag coefficients of uncambered wing section Fig. 57 Drag coefftcients of slightly cambered wing section T /.e

48 upon model open-water tests such as those in Figs. 1S through 26 questionable. would like to ask the authors for a brief outline of how they would modify their design procedure such as contained in their reference [6] to ensure obtaining the best propeller as well as the desired full-scale perfornmnce. Maurice R, Hauschildt, Member: The authors are to be congratulated on a very thorough effort in providing a systematic series of propeller characteristics for use in future design efforts. Both the open water and cavitation tests with the higher blade numbers and larger blade area ratios were sponsored by the half of the Bureau of Ships that has now become the Naval Ship Engineering Center, and the purpose was to increase propeller characteristics information in these areas. The effect of Reynolds number does raise some questions of accuracy with these results. understand that you have a variable-pressure towing basin under construction at NSMB. Do you plan to test these propellers in this new basin with standard ship models, to compare these cavitation results with those for propellers installed on ships? f so, when could the results of this comparison be made available? Are there any plans to develop the same type of information for ducted propellers as has been presented for conventional propellers in this paper? Authors' Closure Before answering each discussor in detail should like to make a general remark with respect to the effect of Reynolds number on the optimum diameter and efficiency. The results presented in Fig. 29 have all been related to a fully developed turbulent flow around the propeller blades. n many cases such a turbulent flow will not occur at propeller models for a Reynolds number lower than f we are aware that most of the open-water tests have been carried out at a Reynolds number of 3.1;" we have to accept that laminar flow effects will lead to higher et~ieieney values than can be expected for turbulent flows. So the differences between the experimental values as have been published up till now and the full-scale values will be considerably smaller than might be derived from Fig. 29. Nevertheless for a Bp-value of 5 and a fullscale diameter of 8 m there still remains a differ- ence of 5 percent in efficiency and 4 to 5 percent in diameter between the up-till-now-published experimental data and the full-scale prediction. We are aware that by using these new results of the Wageningen B-Series, which have all been analyzed for turbulent flow at a Reynolds number of 1(16, we have also to look carefully to scale effect in the other propulsive components such as the wake factor, and thrust deduction in order to get the right overall correlation. Jl~rorga.n and Powell made some remarks about the B,,-value being not dimensionless and asked whether the density o is for fresh or salt water. t is for fresh water. f2owelz and 13jOrne asked why the conventional way of plotting the results of systematic screw series has been maintained. We have selected this conventional way of plotting because this is so familiar to the designer. On the other hand, now that the coefficients of the polynomials of the screw characteristics are known, each specialist can derive from these published coefficients his own favorite way of plotting. 13jarne asked why we kept the rpm constant and varied the pressure in determining the characteristic cavitation curves. He would prefer varying the rpm at constant pressure. n our opinion such a procedure does not make any significant difference. By varying pressure at constant rpm one moves vertically through the cavitation diagram. By varying the rpm at constant pressure one moves horizontally through the diagram. The only important point is that one determines the characteristic cavitation curves at the desinent cavitation points. A small drawback of varying the rpm might even be the varying Reynolds number. Bj/irne's suggestion to analyze the cavitation inception diagram with the equivalent profile method is in our opinion too ambitious. Threedimensional effects and tip vortices would not be treated in a proper way. n reply to Dr. Pien we should like to mention that in our paper the profile data as published by Hoerner have been used. Especially Hoerner's functional relation between the drag coefficient C1) and the flat-plate friction coefficient and the thickness-chord ratio. We are convinced that data as shown by Dr. Pien have been included in Hoerner's summary which has been based on an enormous amount of profile data. To Morgc~n and Cox we like to make the following remarks: 1 The authors agree that in the cavitation diagram the incipient curves are not so accurate as one might desire. This is due to air content and other unknown factors. On the other hand 31 6 The Wageningen B-Screw Series