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On the Optimization, Including Viscosity Effects, of Ship Screw Propellers with Optional End Plates K. de long (University of Groningen, The NetherIands) ABSTRACT For ship screw propellers an optimization theory is discussed, that can be applied to propellers with and without end plates. The method differs from the classical optimization in that for the derivation of the optimum circulation distributions the sum of kinetic and viscous energy loss of the propeller is minimized instead of kinetic energy loss only. A criterion for the risk of cavitation, a simple strength model and an optional nonlinear correction are incorporated in the optimization method. Hydrodynami- cally favourable planform shapes of screw blades and end plates are discussed. Numerical results of optimum screws are given, including a comparison between the application of different types of end plates. It is shown under what circumstances and to what extent the theory predicts a higher optimum efficiency for propellers with certain types of end plates than for propellers that have no end plates. This paper is a summary of [1i, in which some topics are treated at greater length and in which more numerical results are given. 1. INTRODUCTION In [2] the design of a model screw propeller with end plates is discussed, and results of experiments are given. These results are promising for the end plate concept. In this paper we want to obtain a better insight into the circumstances under which optimum propellers with end plates can have a higher efficiency than optimum propellers without end plates. In the classical optimization theory for ship screw propellers, as used for instance in [2], [3], [4] and [5], when the propeller reference surfaces are given the optimum circulation distributions are determined by solving a variational problem such that the kinetic energy loss of the propeller is a minimum. A major subject in this paper will be an optimization theory in which in the variational problem the sum of kinetic and viscous energy loss is minimized. In this paper we will speak of the latter optimization theory as "optimization including viscosity". Bearing in mind the results of [3] we confine ourselves to propellers with zero rake and with end plate planforms lying in a circular cylinder. Furthermore, for reasons of simplicity, we restrict ourselves to propellers with homogeneous inflow. K. de Jong, University of Groningen, Department of Mathematics, P.O.Box 800, 9700 AV Groningen, The Netherlands. 585 Viscous energy loss depends mainly on the distributions of chord length and of maximum profile thickness along the spans of screw blades and, if it does apply, of the end plates. When we want the risk of cavitation to be approximately equal along all spanwise stages of blades and end plates of a screw propeller, the distributions of chord length and maximum profile thickness can not be chosen arbitrarily. Therefore in sections we will introduce a criterion for the risk of cavitation, being a relation between circulation, chord length and maximum profile thickness, depending on some parameters. That the cavitation criterion can be satisfied for a "two-sided" end plate, is discussed in section 4 by explaining the concept of an end plate consisting of two "shifted" parts. The required strength of the propeller also puts constraints upon the distributions of chord length and maximum profile thickness. Therefore an approximative strength calculation is embedded in the optimization method, as is explained in [1~. The relation associated with our cavitation criterion, makes the viscous energy loss of a screw propeller depend in essence only on the circulation distributions of the screw, see section 5. Because it is evident that also the kinetic energy loss depends on the circulation distributions, we are able to formulate in sections a variational problem for the minimization of the sum of kinetic and viscous energy loss, which both can now be considered as functionals of the circulation distributions. The optimization including viscosity seems to be important especially when propellers with end plates are optimized. This we expect because end plates have the effect of relatively increasing the circulation and chord length at large radii with respect to a propeller without end plates, while at large radii the viscous energy loss becomes important. Also for lightly loaded propellers which have a relatively large viscous energy loss the optimization including viscosity seems useful. For more heavily loaded propellers the kinetic loss becomes predominant, while the prediction of the kinetic loss for higher loadings is more uncertain because of the nonlinear character of the problem. Therefore we made a rough attempt to correct the linear theory by iteratively adapting the advance ratio of our vortex wake. Some design requirements, hydrodynamical aspects, strength aspects, aspects of the viscous energy loss, our cavitation criterion and our nonlinear correction

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coordinates (x,r,0) are such that in the yz-plane the angle ~ vanishes at the positive y-axis and increases towards the positive z-axis. The screw has a rotational velocity ~ > 0 (rad/sec) around the x-axis in the negative B-direction. There is an incoming homogeneous flow U in the positive x-direction. In our model we neglect the influence of the ship and of the hub, hence we consider freely moving screw blades. The reason for the neglect of the hub is that then in the optimum case the circulation vanishes at the root of the blade, and it seems reasonable that when such a blade is mounted to a hub of finite length, no strong hub vortex will occur. A more fundamental approach for the hub of finite length is given in [6~. The blade and end plate are situated in a close neighbourhood of their planforms, which are parts of the corresponding reference surfaces. The reference surface Hb of the blade is Hb: 0=~x/(u+vx), rh<~ OCR for page 585
r=rp, a~=a4e r=rh,~6=0 spy ?' ., Fig. 2 Impression of planforms ol (a) x ~ ~ ; (b) suet ton side . O . .- (a) Expanded planform Pb of blade (b) Developed planforms Pp and Ps of end plate at radius r-rp reference surfaces Hi, (i = b, p,s). In Hb the parameter Xb vanishes at the y-axis. In H`, (i = p,s), we reckon Xi zero at the line which is orthogonal to the helices of Hi and which passes through the point R at the y-axis. Chord lengths at blade and end plate are denoted by ci(ai), while the maximum thickness distributions of the profiles are denoted by titai), (i = b, p,s). The quotient titai)/ci(ai) is called the thickness ratio. Each reference surface Hi, (i = b, p,s), has two sides, a "+" side and "-" side, which are defined as follows. At Hb the "+" side corresponds to the pressure side of the screw blade. At Hp the " + " side is the side facing smaller radii, while at Hs the " + " side is the side facing larger radii. The " - " sides are the opposite ones. See Figure 1. Although in Figure 1 for simplicity a one-bladed screw is drawn, we will consider in the following Z-bladed screws with Z> 1. The blades and end plates of a Z-bladed screw all have the same geometry and are equally spaced. This means we have Z sets of planforms Pb, Pp and P9, each set rotated with respect to a neighbouring set over an angle 2,r/Z radians around the x-axis. The use of a second subindex j, (j= 1,...,Z), refers ta the specific number of a blade. For instance for the Z-bladed propeller the reference surfaces are denoted by Hij, (i=b,p,s; j=l,...,Z). We shall make use of the expanded blade area ratio Ae/Ao given by Z P z b~e Ae/AO= r Cb(~) dr=- 2 iCb(ab) dab (4) ~rrp O ., 1 v ~ ~rp ~h (U~ V~ The outer circular cylinder (r=rp) is partly covered by the reference surfaces Hp j and Hs j, (j=1,...,Z), of the end plate parts. The ratio of covering k is defined by k = Z (Bs~ Bp) / (2~) (5) We only consider screws with k in the range 0 < k < 1, because from the potential theoretic point of view k>1 yields the same efficiency as k= 1 while the viscous resistance of the end plate only increases. Splitting k in the contributions of the pressure sides and of the suction sides, we define kp=-Z0p/(2~) , ks=z~s/(2,r) , kp+kS=k . (6) 3. RELATION BEIWEEN CIRC~ULATION, CHORD LENGIH AND MAX[ML~1 PROFILE THICKNE~ In this section we assume the chordwise pressure jump distribution along a profile of a blade or end plate to be prescribed. In design methods often pressure jump distributions are encountered of the type given in Figure 3. The parameter Xi is the length parameter in chordwise direction and ci = citai) is the chord length of the wing section at the spanwise position ai of the planform Pi. Furthermore we denote by [Pi]-(ai,Xi) = P2 (ai,Xi) - p'(ai~xi) , (i = b, p,s) , `7' 587

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the pressure jump between the "+" and "-" side of a wing section per unit of length in the chordwise Xi direction. The parameters 0`i = ot`(a`) and pi = p`(a`) determine the shape of EPIC+, they are taken in the range O < 0`i < pi < 1. t[P,] CPtAi~ 1h ~ ~- ~ /! \ V. I\ ALE XLEto~ C XLEt4IC' NOTE X, Fig. 3 Prescribed chordwise pressure jump distribution of a wing section, O < Hi < ~` < 1. A wing section (profile) at the radial position r has the velocity V`=V`(a`) with respect to the water V(a) ((wr)2+(U+v )2~1/2 (8) where p is the fluid density and where the constant Aita`) equals Alta`)=2/~1+pi(a`)-or`(ai)~ (10) In agreement with the sign definition of ~ and orb, the circulation ['i(a`) is taken positive with a "right-hand screw" in the positive direction of the length parameter ai. The actual pressure at pressure and suction side of a wing (in our case a wing can be a screw blade as well as an end plate part) also depends on the thickness distribution of the wing. For instance, consider a two dimensional flow around a symmetrical profile with zero angle of attack. The fluid flow at infinity is the uniform parallel flow Vitai). When the thickness ratio ti~ai)/ci(ai) is sufficiently small, the greatest drop of pressure is quadratic in Vital) and about proportional to the thickness ratio. Therefore, omitting terms which are of quadratic or higher order in the thickness ratio, we can write pi (ai)-TOO= -p A' (~i) V'(ai) i~ai ~ where the corrective velocity AX will be discussed later on. At the end plate we have r = rp, hence Vitai)=Vb(abe) for all values of Hi, (i=p,s). To create the pressure difference over a planform we assume that the planforms and the reference surfaces downstream of the trailing edges of the planforms are covered with vorticity. To represent the thickness of the blades and end plates we assume an appropriate distribution of sources and sinks over the planforms. The vorticity ~ita`,Xi) has two components. The free vorticity, denoted by /(ai'Xi), is the component in the helicoidal xi-direction, hence it does not deliver a lift. It is reckoned positive with a "right hand screw" in the positive direction of the length parameter X`. The bound vorticity, denoted by a), is the component perpendicular to the helices in a planform Pi, hence it isb the pressure jump generating component. The sign of yi is taken equal to the sign of the corresponding pressure jump [Pi]-. Both min ~ components ~ and yi are taken per unit of length in Pi (ai)-Poo=PilAifai)~ their perpendicular directions in the reference surfaces. Note that these perpendicular directions at Hb are not the coordinate directions of the (ab,Xb) system, which is not orthogonal. At the reference surfaces Hp and Hs belonging to the end plate however the perpendicular directions are the same as the coordinate directions (a`,Xi), (i = p,s). We remark that we call free vorticity that component of the vorticity which does not give rise to a pressure jump, even when it is situated at the planform of blade or end plate. In our theory the picture of the chordwise pressure jump along a wing section (Figure 3) is proportional to the picture of the bound vorticity yb(a`,X'). Hence the prescribed pressure jump distribution determines how the circulation [` = lipid`) around a profile is distributed as bound vorticity (ai'Xi) along the chord. This means that when F`(a`) and octal) and Midair are fixed for a specific wing section (that is i and Hi are specified), the pressure jump is inversely proportional to the chord length ci(ai), hence max ~ [P`~-(ai,X`) ~ = o OCR for page 585
use of an adequate weight function wb=wb(ab) and we suggest the following relation between circulation and chord length ci(ai)=Bwi {~ A' Vi ~T`~ +A' V' ti} lo _ h of ol oft.` 1 {1~\ where we introduced the constant B defined by B p/(pmin p ) If we had considered a variable static pressure and hence a variable ambient pressure pa, then from the for cavitation most dangerous vertical upward position of a screw blade, we could derive that it is better to use functions Bi=Bi(ai) instead of the constant B. However, as remarked, for simplicity this is not considered. The way in which we choose the weight function Wb - wbtab) is a subject which is discussed in [13. It is taken into account in the optimization method, which determines among others the functions Pi = [i(ai) and ti = titai), (i = b,p,s), occurring in relation (13), and its choice has an influence on the determination of these functions. In relation (13) we also introduced for the end plate the weight functions wp=wpfap) and wS=wstas). We introduced the weight functions wi as a tool which can be of use in the optimization method, to give weight not only to the constraint related to the cavitation danger, but also to constraints concerning hydrodynamics, strength and other aspects, see [1~. 4. DISTRIBUTION OF VORTICITY AT THE REFERENCE SURFACES We assumed already that the vorticity belonging to the lifting surfaces of blades and end plates, is lying at the planforms P.`, (Figure 2). Because the vorticity field is free of divergence, we can define at each planform Pi and its corresponding reference surface Hi a "vorticity stream function" ~` = lPi~a`,x`), (i = b, p,s). The meaning of such a stream function is the following. For two points (cri,X`) and (a`,X`), both situated at Hi, we have (~i,xi) ~ita`,Xi) - ~ikai,X`) = Jl An ds , (i = b, p,s) , (15) ( ~i,x`) where Tn is the component of the vorticity y perpendicular to the line element ds of an arbitrary line connecting the two points The leading edges, X`=Xi (a`), of the planforms Pi are "stream lines" of~kthe vorticity field. Upstream of the planforms, (Xi < Xi ), there is no vorticity. Hence we can choose ~i(ai,Xi)-O , Xi Xi by the special case of (15) xi Mimi xi) = J r2tai~xi) dxi xi>xi fail ~ (18) X[E( Hi) where we used that hi is taken zero at the leading edge, see (16~. By (15~-~18) the stream functions Hi are uniquely determined at the two-sided infinitely long reference surfaces. Because of the preservation of the vorticity stream we obtain a picture of the vorticity lines by plotting the level lines of the functions ~i. In the case when an end plate is mounted to the tip of the screw blade we discuss the important issue of how to convey adequately the vorticity of the screw blade to the end plate. It is desirable that the vorticity at the lifting surfaces and the trailing vorticity do not induce too large (or theoretically infinite ) velocities at the lifting surfaces themselves. Otherwise, from the view point of the vortex theory, the desired propeller could not or only unrealistically be constructed using a lifting surface theory. A first consequence of the above is that in the vortex model of the screw, it is desirable that no . . . . . concentrated vortex line segment occurs at the propeller, because, following the law of Blot and Savart, such segment would induce at the propeller velocities which are inversely proportional to the distance from the segment. The possible occurrence and the avoidance of such a singularity will be discussed later on. A second consequence is that discontinuities in the strength of the vortex sheets, inducing infinite velocities at the planforms, have to be avoided as much as possible. We illustrate this point as follows. Consider, (Figure 4), in the half planes I and II, two vortex sheets of constant finite strength yI and rII respectively, with yI ~ rlI. The vorticity is parallel to the straight line 1, at which the two sheets are connected. XiTE(~i ) ri(a2~= Jo )~(a`,Xi) dXi , OCR for page 585
line I according to a singularity which is logarithmic in the distance to the line 1. This second consequence implies for instance that at leading and trailing edges of a planform Pi the induced velocity would be infinite, when we had chosen 0`` = 0 and Hi = 1 in the chordwise bound vorticity distribution given in Figures. Hence from this point of view values of Hi and pi with `i>0 and p`<1 are to be preferred. To illustrate some basic ideas we direct our attention in the remainder of this section to a screw with a two-sided end plate of which both end plate parts have equal span a ,e = as,e or kp = ks. When the functions Job' Cb, `b and i3b are given as in Figure 5(a), (b) and (c) respectively, we can draw pictures of the vorticity field in the expanded screw blade planform by using (1S)-~18). In Figure 5(d), (e) and (f) for different choices of leading edges of the screw blade, results are drawn. We remark that the motivation for the choice of the shape of the functions fib, Cb, Cab and fib iS given 3.2 - b to A/// , ol /- 0 Pb (m2/sec) ~ 17 0 cat (m) (a) \\\\\\\\ ~ ~ LE in later sections. By ASP`, (i = b, p,s), we denote the difference in the stream function Hi between the leading edge and the nearest depicted vorticity line and between each two depicted neighbouring vorticity lines. We must realize that the density of the vorticity lines in Figure 5(d), (e) and (f) is not an exact picture of the density of the vorticity lines in space, because the surface Hb is not flat and even not developable. However a good insight can be obtained in this way. For the three types of considered screw blade planforms it is seen that at the blade tip there are ranges of Xb in which the free vorticity strength does not vanish /(ab,e,Xb) ~ O ~ Xb ab,e, hence at the radially lengthened blade planform Pa, we know that there is no vorticity, and hence ~ = 0 for ab > ab e, we conclude that a 3.2 \\\\\\ LE In_ 3.21 `b | lab <~ _ ~ Fig. 5 Vorticity distributions at three different, expanded screw blade planforms Pb of a propeller with end plates; /\~b = 0-7 m2/sec. 590

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discontinuity arises of the type given in Figure 4. It satisfies is observed that this type of discontinuity is less strongly in the case of Figures(d) than in the case of Figures 5(e) and (f). Now we come to the question as to what kind of two-sided end plates have to be mounted to screw blades whose planforms are for instance as depicted in Figure 5. In order to avoid the creation of a concentrated vortex line segment at the junction of blade and end plate planforms, it is necessary to choose the bound vorticity of the end plate parts such that at the line ap_ag=O, hence at the roots of these parts, their sum equals the bound vorticity at the tip of the screw blade rp( ,xp ) + As ( O. AXE ) = ~b( lab, e'Xb ) ~ Xb ((7b,e) r, (m2/sec j - b" ~ ,0.6m 0.6 r, ( In lace ) b ~ - 1 17 (a) (d) Orb orb '\ (e) | (up = 0) rat ' (f) (g) 0`p(0) = 0`s(0) = 0/b(ab,e) I ~p(O) = 8(0) = ~b(ab,e) . (21) An end plate of this kind was used for a model screw propeller as discussed in [2]. An example of such an end plate is given in Figure 17. For this type of end plate we have no concentrated vortex line segment at the junction of blade and end plate and because the free vorticity at the two symmetric end plate parts is of equal strength at their connection, also there exists no discontinuity in the vorticity field of the type of Figure 4. Aside we remark that experiments in a cavitation tunnel showed that the more or less concentrated blade tip vortex, which normally arises for propellers without end plates that have a square root singularity in the circulation distribution at the blade tip (see for instance Figure 14) could not be made visible for the model screw propeller with this type of end plates. The reason is that the trailing vorticity is spread over the trailing edge of the end cp (m) '' ~ . ~7 (b) (ab = able) ~.. . . ~ /' ' 1\ _ _. ~ ~ A B E F 0.6 m _ . P P . _ b t O \ CYp W~ UP / \ I b ~~ it, _ O.8m ore ;~, (t ) 1 2 / (B V5(abte)) ~i\ ~ / I ~ \ ! 2 / (B Vb(ab,e)) / 1 .... 1 ~ ~ C D E F A B ~ A ~ ~ B i\ (as= o) C D c HE Fig. ~ Vorticity distribution belonging to a two-sided end plate with planforms which are relatively shifted with respect to each other; ~Pp=~P,=0.7m2/sec. 591 2 / (B Vb(ab,e) )

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plate, compare Figures 14 and 17. However a disadvantage of the above end plate is that its chord lengths are relatively too large compared with the chord length of the screw blade. Since only half of the circulation of the blade tip is conveyed to each end plate part, in relation with the avoidance of cavitation only about half of the chord length of the blade tip would suffice for the chord lengths at the roots of the end plate halves, see relation ( 13 ). Obviously, shorter chord lengths will cause smaller viscous losses, and min~rnization of the viscous loss especially is important for the end plates, because they have a relatively large velocity with respect to the water. This will be discussed now. For simplicity we choose in first instance chord lengths based on relation ( 13 ) without taking into account the thickness. Furthermore at the end plate near the blade tip (0 OCR for page 585
x=cons t +U+vx Ekin = Y2p JJI Igrad~(x,y,z) l2dxdydz , (24) x=const . where ~ = ~(x,y,z) is the velocity potential of the trailing vorticity lying in the two-sided infinitely long reference surfaces Hi j (i= b,p,s; j= 1,...,Z). It is a straightforward task to reduce equation (24~) to fib , e E~in(ri) =-v2pZ ~ J ~nbrb(ab)Vb(ab)dab + o or i ~ e Vb(ab,e) ~I ~Uiri(ai)dai1 ~(25) where ni (i=b,p,s) is the unit normal at the reference surfaces Hi', directed from the "-" to the "+" side. When the screw blades and the end plates move through the water, frictional forces arise. Consider across a wing from leading to trailing edge an elementary wing strip of width dad. The strip has the chord length ci(ai) and maximum thickness t`(ai) and is moving with relative velocity V2(ai) through the water. For the resultant viscous force dFt2 ~C(a`) on the strip we use the formula drip (hi)= 2 p (at) Vt(ai) ci(ai) dai , (26) where the section drag coefficient C7(ai) is defined by C7(ai)=2C:(a`) (l+A' (a`)titai)/cifa`~) (27) In expression (27) there occurs the skin friction drag coefficient C:(a`), for which we take C:(a`~=0.075 ~ logReyi(ai)-2)~2 . (28) The constant AII (Al) in a section depends on the location of maximum thickness of the wing section. For instance for sections with maximum thickness located at or near 30% of the chord, Ai-Ii can approximately be taken as 2.0, however for sections where this location is at 40 or 50% of the chord, the value 1.2 will be more appropriate, see A. The local Reynolds number Reyi(ai) is taken as Reyi(ai) =Vi(ai) ci(ai) / v , (29) with in the denominator the kinematic viscosity of water ~ = 1.2 10-6 m2/sec. The approximative formula (26) should not be used for Reyi below 5*105. Further we remark that the drag, coefficient C7, (27), depends on the lift coefficient C: of the wing section, in that at higher lift coefficients, a greater part of the drag is contributed by pressure or form drag resulting from separation of the flow from the profile. See for instance [83. One can think for example, for some types of profiles of the "bucket-shaped" drag minimum of the function C3~=C7(C:). For simplicity we did not include this effect in (27), however in the iteration method that we discuss in section 7, one can account for this influence on C: in step 5 of the iteration scheme of Figure 7. The work done by the frictional forces causes energy loss. This energy loss per unit of time we denote by EWC Denoting the contribution of the wing strip to this energy by dE:t~C(ai), we have dF:i8C(a`)=Vi(ai) dF:i8C(ai) , (i=b,p,s) . (30) Using (26) and (27), choosing the chord length ci = ci(a`) according to relation (13), and taking the sum of the contributions of all Z blades and end plates, we can write EWC as a function of the circulation distributions F`, (i=b,p,s), Ire EWC([it)=PZ i, 2 ~ C:AtI! V3 tidal ~ _ i=b, pa, O i, e BJ~V2 Wi I - t |Ti| +Ai Vi ti} daiJ (31) o The dependence of EWC on the circulation distributions is a complicated one, when we realize that the skin friction drag coefficient C: depends via the Reynolds number Reyi and the chord length ci on the circulation loci, see formulae (28), (29) and (13). The component of the frictional forces in the direction of the screw axis counteracts the propeller thrust. Using (26) and again relation (13) we find the resultant thrust deduction TIC due to viscosity bite TWC([i)=P(U+Vx)Z ~ J HI ~Vitid~i + i =b, pa, lo. ',e B i~ Vt hi { hat |ri| +A' Vi ti} deli . (32) o For the potential theoretical thrust Tpot we take Tpot=P~Z J ~(~b)rb(~b)dab ~( ) o whereby the thrust T becomes T= 1 pot-1 DISC (34) Finally we remark that in the approximation of the viscous energy loss ERIC and of the thrust deduction due to viscosity TIC we have neglected the positive or negative interference drag produced in the comers of the blade-end plate junctions. The interference drag is caused by the interaction of the boundary layers of blades and end plates and it depends among others on the thickness ratios ti/ci alla on the lift coefficient CL. Of the wing sections near the junction. 6. FORMtL\TION AND SOLUTION OF A VARIATIONAL PROBLEM In our screw propeller model the energy-balance equation reads Q(ri) ~ = (T',ot(ri)-Tvisc(ri)) U+Ekin(ri)+Evisc(ri), (35) where Q(ri) is the torque about the propeller axis and Tpot' Tvisc' Ekin and Evisc' as functions of the 593

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circulation distributions pi, are given in (33), (32), (25) and (31) respectively. Evidently it is demanded that Tpo`(Ti) > T~,C(ri). For the propeller efficiency we have useful work (Tpot([ i ) - T2,, ~c(ri) ) U total work Q( r i )~ In this section we assume rh/rp, U. ox, Z. rp, I), kp and k,, to be given, that is the geometry of the reference surfaces is given. We minimize the sum of the kinetic and viscous losses, (Ekin+Evgac)' under a number of constraints. One constraint is that a prescribed thrust T has to be delivered Tpo~(Ti) -TV) = T . ( ) Notice that when we place a bar on top of a symbol, we want to stress the fact that the variable in question is prescribed. Another constraint is that the distribution of circulation and of the chord length along the span of blades and end plates has to be such that the danger of cavitation is about the same for blades and end plates. It can be demanded that along the spans of blades and end plates the chordwise magnum pressure at the suction side of blade or end Slate equals a prescribed minimum pressure level porn = p Ant or equivalently B = B. see (14~. Instead of this last condition, another one can be required, namely that the screw has a specific blade area ratio Ae/Ao, (4~. Thus we formulate two problems. Problem I: We assume T=T and B=B to be prescribed. Then we want to find the optimum circulation distributions PiPt (i = b, p,s), belonging to the given quantities, from which follow automatically the optimum chord length distributions coins by relation (13~. Hence for PiPt we require the efficiency A, (36), to be maximum and therefore (E~(Pi, )+E~ec(Pi )) to be minimum, under the constraint that 15 satisfies (37). Then the blade area ratio Ae/Ao follows from the optimum chord length distributions C.Pt by formula (4). Problem II: We assume the thrust T= T and the blade area ratio Ae/Ao = Ae/Ao to be prescribed. Then it is possible to calculate the corresponding minimum pressure level porn in the optimum case. We remark that problemII can be important for the purpose of tuning the constant B in our cavitation criterion on the basis of existing screw propeller designs, for which the blade areas are known. For both problems we consider the functional J=J(Il`) given by J(I`i) =E`c`.,([i) +Ev='c(l~i) -A (Tpo`(~i) -Tv~c([i) - T) , (i = b, p,s) , (38) where ~ is a Lagrange multiplier. For a functional J=J(I,i), (i=b,p,s), we denote the Gateaux-variation at [i with respect to a set of perturbational functions hi, (i=b,p,s), by [J([i; hi) The Gateaux-variation, also called the first variation, is given by ~J([i; hi)=E-m (J(IIi+~hi)-J(~i))/6 , provided that the limit exists. Here we remark that in our model rift vanishes automatically at the root and the tip of the blade when the screw propeller has no end plates (ratio of covering k=0~. Otherwise a concentrated hub vortex or tip vortex would arise which theoretically gives an infinite energy loss. For the same reason, when the screw propeller does have end plates, I:Pt has to vanish at the end plate tips when 0 OCR for page 585
if=) (~1+B (P2)+B dp3 To satisfy the condition of vanishing total circulation around the free vortex sheets, we require each of the three potentials to be uniquely valued in the whole space, their boundary conditions are Hb ~ 1 = _~) , Hi : aft =0 , (45) .=p,8 (i-b p a) Mini W(U+vx)C:ViwiA' , (46) Hi : [~33473 = hi C Vi3 wi A, ( i=b,p, s) , (47) where the potential Al is the optimum potential belonging to the optimization in which only kinetic energy loss is minimized. For the boundary value problems (45), (46) and (47) some properties of the corresponding free vorticity strength in the reference surfaces if`, (i = b, p,s), infinitely far behind the propeller are discussed in [1~. These properties are necessary to formulate wellposed systems of singular integral equations which are equivalent to the boundary value problems and which can be solved numerically by a collocation method analogous to [3]. In the beginning of this section we formulated two problems, problem I and problem II, to be solved. Assuming we have solved numerically the circulation distributions [!~]_(a`), [~2]_(ai) and [~33_(ai), (i= b, p,s), we now explain how the resulting optimum circulation distributions Pi (of)= { [~l]_(ai)+B Eqi2]-(~i) }+B [~33_(ai), (48) are obtained for each of the two problems. In the case of problem I where we have a prescribed minimum pressure level pmin= pmin we know by formula (14) the value of B= B in (44). Then the Lagrange multiplier ~ can be solved from the prescribed thrust T= T by substituting (44) in (34) with the use of (32), (33) and relation (13). We find ~ _ ~ T/(pZ) + K - B J(4P3) ~ _ J(4P1 + B 4~2) where K and Jo) are given by ~(49) al e K = (U+vx) >, J C`FVit`(A``Ii + BVi220iA2I ) dai , (so) i=b, O P,8 Ja e J ( ( / ~ ) = { ( ~ ) r - ~ 2 B ( U + v x ) c F ~ v i 2 o i A ~ } [ ~ ] + d a b o a. -~B(U+vx) ~ j-C,FVi22eiA~[~]+ dai . (51) i=p,8 0 For Problem II we have to solve ~ and B from the prescribed thrust T= T and the prescribed blade area ratio Ae/Ao=Ae/Ao. This gives us two equations for the unknowns ~ and B. one by substituting (44) in (34), the other by substituting (44) in (4), using for the chord length relation (13). The more complicated formulae that arise for the solution of ~ and B are skipped here, but are given in [1~. 7. II~RAlIVE DEI=MINA1ION OF SOME: DESIGN REQU~ In this section we discuss how some design parameters and functions, occurring in the solution of the variational problem discussed in section 6, are iteratively determined. _ _ When rh/rp, U. Z. rp, a, Up, k`,, T= T and (B_ B or A~/Ao = Ae/Ao ) are given, an iteration method can be carried out according to the scheme given in Figure 7. Prescribed quantities: rh/rp, a, z, rp, is, kp, k`,, T=T, l Problem I: B = B , Problem II: Ae/Ao = AJAo Iteration scheme: - 1 initialize :vx = 0, :'D`(a`)_1 :Reyi(a`)_l*lO6, :ti(a`)_lower bounds, : a`` ( Hi )-O. OS, pi ( al ) _0. 8 2 |calculate ~ :[0l]+(ai) from conditions (45) . 1 3 calculate :[~2]+(a`) from (46), +(a) from (47) . ~ , 4 determine :A (ProblemI) :B alla ~ (Problem II) 5 correct :,u`(a`), :Reyi(ai), C:(a`), C7(ai), :ci(ai), ti(ai) :c~i(a`), fitai), A,(~i) if the corrections are still significant, then else |correct Vx=~. | if vx still changes to some extent, then 1 1 Fig. 7 Iteration scheme for the optimization including viscosity when the given quantities are prescribed. In the initialization slept of the iteration we introduced lower bounds for the maximum thickness distributions ti = t`(cri). As the lower bound at the screw blade we take 2 rp f th, where we introduced the 595

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thickness factor [th. At an end plate part we choose as lower bound for the maximum profile thickness distribution, a tapered distribution from the) = 2 rp Ah at the end plate root to ti(ai,e) = rp fth at the end plate tip, (i=p,s). Thus the use of the lower bound in particular results in a nonhero thickness at the free ending tips of blades and end plates and according to relation (13) also in a nonzero chord length there. In the calculations used in this paper we have used the value fth=0.0035, see (52). For the calculations in step 2 and step 3 of the iteration we use the collocation method described in [33. When we want to apply only the linearized theory, then in the iteration we keep ox equal to zero, and hence leave the outer loop undone. Step4 is the determination of the Lagrange multiplier ~ from (49) (Problem I), or of ~ and the constant B belonging to our cavitation criterion from more complicated algebraic formulas (Problem II ), see [1~. In steps the occurring functions are adapted according to some design aspects, which are discussed more closely in t13. For instance a strength calculation based upon simple beam theory is incorporated in this step. In the strength calculation the considered force fields acting on the beams representing the screw blades and end plates, are the potential theoretically induced lift forces, the centrifugal forces and the viscous forces. Another adaptation concerns appropriate choices from a hydrodyn~mical point of view of the functions ~xi=cxi(oi) and ,B' = pi(ai). For some different types of screw - propellers values of the occurring functions which are eventually found using our iteration method, are given in Figures 14-17. Finally in step6 we want the velocity ox to be such that it equals the Lagrange multiplier A, multiplied by a factor ~ with 6:~. The reason that we take ox = 6A, is that in the linearized lifting line theory where only kinetic energy loss is nnnm~ized, the induced backwards translational velocity of the vortex sheets infinitely far behind a propeller, equals the Lagrange multiplier A. Then in the neighbourhood of the blades and end plates of the propeller the induced velocity is about ~ in the positive x-direction. For higher thrusts the optimization including viscosity, resembles the kinetic optimization more, because in that case the kinetic loss becomes relatively more important than the viscous loss. Therefore we expect that the corrective velocity vx=~A, which has a greater influence for higher thrusts, will be useful of the optimization including viscosity also. Here we emphasize that it is not claimed that the use of the corrective velocity ox gives an exact result for the kinetic energy loss for higher thrusts. We did not take into account the vortex sheet deformation, and the induced velocity by the vorticity and the source-sink distributions representing blades and end plates. However the influence of the trailing vorticity, which delivers a substantial part of the induced velocities, is possibly treated more realistic than in the pure linearized theory. 8. SOME ASI~IS OF l - ; OPTIMIZATION MEIlIOD Preliminary choices In this section we first make choices of some design parameters and functions that are kept the same for the calculations in this paper. Of course these particular choices fire- not essential for our optimization method. \Ve take P = Pseawa~er = 1023 kg/m ~ Pacrew = 7650 kg/m3 rh/rp = 0.2 , ~ = 6 m/see ~ (tlC)h = 0~2 {b( 3(Jb,e) = 0~05 , Ab( 3-~b,e) = 0.8 ~ fth = 0.0035 , SPer = 5.6*107 N/m2 for Problems: B=0.024 sec2/m2, (52) where SPer is the prescribed permissible stress level for use in the strength calculation. The density of the material of the screw P'cr~' and the prescribed permissible stress level SPer are chosen corresponding to the material cunial bronze. SPer is based on load variations of 50 %. We choose for blades and end plates chordwise thickness distributions belonging to the NACA16-series sections (see for instance [83), which for ship screw blades are commonly used. The location of maximum thickness of the section is at 50% of the chord length from the leading edge, so that for the constant A`l`(ai), introduced in the section drag coefficient C7(ai), (27), we will choose the value 1.2. The symmetrical section at zero lift has its minimum pressure located at 60% of the chord from the leading edge, and from the "basic thickness formsn tables given in [8] it can be derived that the constant A'I(a`), introduced in relation (13) can approximately be taken as 1.25. When, in our calculations, propellers with end plates are considered, we assume the two-sided shifted end plate with the anterior end plate part located at the suction side of the screw blade, unless we explicitly state that another type of end plate is considered. Comparison with the classical optimization In the classical screw propeller optimization, as used for instance in [2], [3], [4] and [5], circulation distributions are derived from a minimization of only the kinetic energy loss E,Cin of a screw propeller under the constraint that the potential theoretical thrust T. equals a prescribed thrust T. Hence in the classical theory instead of our functional J = J(l~i), (38), the functional J~,u,=J~.~,(Ti) given by ]~i)=Etin(~i) - 11 (Tpo`(Ti)-T), (i=b,p,s), (53) is considered, where ~ is the Lagrange multiplier. By demanding the first variation of JC`~(li) to vanish we obtain the conditions Job Vb ~ Hi Toni = 0 ~ (i = p,S) ~ (54) in analogy with the derivation of (41) and (42). We put ~=~1 , {55) so that if, is the potential already introduced in relation (44), and which was solved from condition (45). We want to obtain a comparison between the 596

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optimization including viscosity and the classical optimization. Therefore we use for the classical theory the relation between circulation and chord length which is explained in sections and we derive our design requirements by using the iteration scheme discussed in section7. For the classical optimization as well as for our optimization including viscosity we solve problernI, hence wit_ a prescribed thrust T=T and a prescribed value B=B. Then we derive the value of the Lagrange multiplier ~ from the condition Tq' ~ pot-1 vise = 1 7 which, for a fair comparison, contains the thrust deduction T2,~,C due to viscosity. Analogous to the derivation in section 6 we find from (56) that ~ can be solved from T/(pZ) +K ~ , J ( ~1 ) with K and Jo) given in (50) and (51) respectively. Now we give some numerical results of the comparison between the optimization including viscosity and the classical optimization theory. Note that, strictly speaking, we do not make a comparison with the exact classical theory, because we make the comparison with the classical theory embedded in our numerical iteration method, in which we incorporated various nonclassical aspects. In both theories we use the linearized versions (ox = 0), and we solve Problem I introduced in section 6. We consider propellers satisfying Z=3, rp = 4 m , ~ = 6 ,~ad/sec , (58) for a case without end plates (k=0) and a case with end plates (k=2kp=2k`,=0.5). Results are given in Tablel for three different values of the prescribed thrust T=T for which the dimensionless thrust coefficient CT defined by CT= T / (I* p U2 or rp) attains the values 1, 2 and 3. We remark that, for the optirr~ation including viscosity as well as for the classical optimization, for the efficiency ~ we used relation (36). Table 1 Comparison between optimization including viscosity and classical optimization for propellers without end plates (k=0) and propellers with end plates (k=2kp=2kS=0.5~; linear theory (Vx=o). (56) k Ekin ( 106 Nm/sec) . 0 2.369 9.445 121.23 2.368 9.444 21.23 _ 0. ~ 1. 932 7. 644 17. 16 1 2 3 (57) Evil (10 Nm/sec) 2.871 4.552 6.114 2.895 4.569 6. 129 4.640 7.447 10.02 4.888 7.630 10. 17 1 2 - 3 C.~. . '0(0 . . 67.65 152.87 143.27 67.64152~87 143.21 69.86 56.97 47.85 69.79 56.95 47.84 . 2 3 Optimization | incl. vise. | classical _ | incl. vise. classical From Table 1 it is seen that in the classical theory the kinetic energy loss is slightly smaller than in our theory, however the sum of kinetic and viscous loss is larger, resulting in a lower efficiency. To obtain some insight into how the difference between optimization including viscosity and classical optimization affects in our theory the corresponding - screw propellers, we give in Figure 8 for the propellers of Tablet with k=0.5 the corresponding circulation distributions. It is seen from these figures that in the linearized theory for the optimization including viscosity there occur somewhat smaller values of circulation at large radii than for the classical optimization. Smaller values of circulation in our theory imply smaller values of chord length, see relation (13). Obviously the cause of the occurrence at large radii of smaller chord length for the optimization including viscosity, is that at large radii the relative velocity of the wing sections is 59) large and therefore the viscous energy loss is 25 25 . ~ at= _ ~ O 0 ab (m) 3.2 0.51ap (m) O (a) (b) rs (m2/sec ) . --hi as (m)> 051 Fig. 8 Distributions of circulation Pi (i-byp,s) of screw propellers with end plates (k = 2kp = 2kS = 0.5); linearized theory (AX = 0) classical optimization optimization including viscosity 597

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relatively important there, while the classical optimization does not worry about large chord lengths. This also explains why the difference between classical optimization and optimization including viscosity is greater for propellers with end plates than for propellers without end plates, see Table 1. Since kinetic energy loss becomes relatively more important for propellers with higher loadings, the difference in the results obtained from the two optimization methods is relatively smaller for larger values of the thrust coefficient CT than for smaller values of CT, as can be seen from Table 1 and Figure 8. Summarizing we can state that the results of both optimization theories differ only slightly. When no solution is found In section 6 in the paragraph following relation (42) we have assumed that our circulation distributions are positive along all spans and therefore we replaced the quotients l~ifai)/~l~i(ai)~, (i=b,p,s) by the value one. When the circulation distributions are numerically calculated we can verify whether this assumption is correct. For pTOblet?! I and problem II there exists a region in the (w,rp,k)-space, for which our formulation of the variational problem does not lead to a solution of the optimization including viscosity. This appears for large values of a, rp and k. For these cases we numerically find only circulation distributions which change sign along the span of a screw blade or end plate, and which therefore are not solutions of our problem. For instance for a screw propeller without end plates (k = 0) the numerically calculated circulation distributions along the screw blade can be as in Figure 9. It is understandable that this type of circulation distribution is found, because since the quotient l~i(ai)/~Ti(a`)~ has unjustly been taken equal to one, at the spanwise stages of negative circulation Pb there can occur viscous energy gain EWC and thrust production TVUIC due to viscosity, instead of viscous energy loss and thrust deductum due to viscosity. rblL t ab c Fig. 9 Example of circulation distribution of a screw propeller without end plates, that is found when the assumption ri(ai)/lri(ai)l=1 is incorrect. It seems that, to solve this problem, another variational approach for the optimization, including viscosity effects, has to be undertaken. In this paper this will not be carried out and in sections we will designate the regions in which our opti~r~ation method does not give a solution. Fortunately it seems that the relevant optima are found in those regions where our method does give a solution. 9. SOME RESULTS OF OPTIMIZATION Efficiency ~ as a function of T. Z. is, up and k in some theories To obtain some feeling for the dependency of the efficiency ,7 on the various parameters we have drawn in this section for some cases pictures with level lines of the optimum efficiency q. As for all calculations in this paper, we made choices from (52). The equiefficiency lines are constructed by calculating on equidistant 21*21 grids of (w,rp), (sA),k) or (rp,k) values, propellers which all are optimized by one of the methods discussed in this paper. Although in reality the level lines of the propeller efficiency are smooth lines, they are sometimes drawn less smoothly. This is due to the discretization on the equidistant grids. The efficiency '7 as a function of two considered variables in each picture has not more than one local extreme. The location at which the absolute maxims of the efficiency is attained is designated by an asterisk in the corresponding pictures. In the pictures on the (w,k) and (rp,k) grids we have designated by the little ball ~ the location of the maximum efficiency ~ of an optimum screw propeller without end plates (k = 0). Notice that the propellers ~ and ~ are in a sense optima of optimum propellers, because all the propellers for which the equiefficiency lines are drawn have optimum distributions of circulation, chord length and the other relevant functions occurring in our model. By ~7 we denote the difference in the efficiency 7' between propeller ~ and its nearest level line and between each two neighbouring level lines, so that from the value of ~ at the point ~ the efficiency it at each depicted level line can be derived. ~ the pictures the ratio of covering k is chosen in the range 0 OCR for page 585
a) 11 - Go 11 i: 11 o ~ a, ~ 11 - ~ o ~ ._ a,) 4= o I ~ 4= o ~ o . - ~, 3 ~ sot of 3 ~ ~ an ~ a c,,, 11 o ~ ~ ._ _ ~ loo 41) al 0 11 o ~4 ~ ///~,,~,,'.' ~+': Co ~3 t3 _ - 4= ._ U) o .U) ~ do 11 X _ id ~ _ ;- . - o c: o ._. N ~ .~ a' ~ _ .= O O ~ ~ ^ o Go - 11 X ~ A ,_ _ .o U) ~ u a, ~ C - ._ _ - .^ - 599 . ___ ~ ~ ~ O O O O _ O E d4 ~ ~ E -O ~ ~ ~ ~ ~* * * ~ 3 ~ 1= ~ ~ o L: ~ 4= ._ o ._ ~0 _ .= O 11 _ X ~a - .._ O O ~ a) ~ S .~ . - ~ - O _

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, 4= ~ - cn o no 11 ~ - :' Is ~ o a) c: =: o ~ ~ lo ~-~ -4 ~ ~ - ~ - C:: ~ o o ~ qu - - 3 ._ UD o v U) .~ he ~ .- o ~ 11 _ X V ~ c: - ._ o o --~ A ._ a) o _ 600 ~ 11 ~ ~- 3 >, 11 o ~ .d t_ o o a, 4= o o o ._ I:: a) ~ i= u, 3 ,% Cal :^ 11 ~ ~ o t~ be 11 ~ ~ ,, .~ ~ 2 - _ ~ o o * ~ o Cal I:] ~ 11 'd ~4 -

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G: / / ~l ~ - - / I 4= ~ - o At >] to 11 x ~-~ ~ ~ - At ~ o a) o ~ .-~ ~ a) ~ - 4= ~ Q O O c: 601 x . ~ .m ~ U] _' o lo, o ~ =, o 4= - o .-o. o Cal U =: 11 cx3 c: ~ ~ s.~Oo as, Cal ~ ~ 11 If .o ~ 11 .Q Cal ~ 11 _ a)~ t-, To 11 o .= ~ :> C) <1, no ~ a) o Co Ha- - 11 Cal bO I ho .= o _ ~ C) ~ ._ o o 4= ~ ._ .S Q `: O _

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From Figures 10-12 we conclude that, in the regions where the optimization including viscosity gives an answer, the efficiency obtained by the optimization including viscosity is quite the same as the efficiency obtained by the classical optimization. The nonlinear v-d-correction of Figure lO(c) gives values which differ not much from the values of the linear theories of Figures lO(a) and (b). The reason is that the propellers have very large propeller radii rp (25m OCR for page 585
1 3.2 _ n ~ ~ O ~ O tb (m)- 0.4 0 r. (m2/sec) _ :1 2 r E b .ol _ 170 Utb ~ 3 3 ~ Fig. 14 The optimum propeller of Figure ll(a); without end plates; ~b = 0-7 m2/sec. 1111/////~1 ~ ~b 0.9. 2! t,(m)* 0.9 3.2 E b 0.9~ \ ~ O _ o.4 ~1 ~ /~/ 0.9 v -O ~101 --- 101\ , I tb (m)- 0.4 0 rb (tn2/sec)- 17 0 ~tb ~ 3 0 `bi~b * 1 Fig.15 The optimum propeller ~ of Figurell(a); two-sided shifted end plate2s; anterior end plate parts at suction sides of screw blades; a~b=a~p=a~s=0.7m /sec. o.9 ~ \ rp (m2/s,ec} ~ _ - ~ o ~ / r, (m2/sec) 17 ^1 ~ ~1 0.9 ~ ~ ! C~P -~13 O I \ ~ c~ I, 3 2 3.2 J.;t O~, tm) _ 0.4 0 r, (m2/sec)- 17 0 to. _ 3 n, . u 1 32T \ t \ v L O / O t. (m) ~ 0.4 0 r. (m21sec) _ - - o 1 .q ~ ~.~ Io , ol 17 0 W.- 3 Fig. 18 The opti~num propeller ~ of Table4 with one-sided end plates at of screw blades; a~b = A~3 = 0. 7 m2/sec. c~ ~ 32T 0.6 01~\ b] 1/ 0.6 3.2 O .~ \~ O O tb(tn) _ 0.1 0 0.6 _ ,4 0 ~> ~L: ol / r, (m2/sec) 1 0.6 0.6 . 3.2 _ ~t O __ * 17 0 , / b t 7112/ ~rc ~ Fig. 17 The optimum propeller ~ of Table 4 with two-sided symmetrical a~Pb = A~p = A~3 = 0.7 m2/sec. 603 ~,;~. ~ / /~.', / u 3 0 C'b;l~b ~ sides ~, O. i ~. I 4,, ' ! 97b ~b 'lb- I end plates; 1 1

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I. al e : : . . : : : ~: Ma ~TE ~a, I, .... al = Glib : : ~ cb(a~e) .. Fig. 13 Leading and trailing edge and corresponding chord lengths of our two-sided symmetrical end plate planform; kp-kS=YJc; i=p,s. plate. Assuming the span of an end plate part to be smaller than the chord length of the screw blade tip, the chord lengths of the two-sided symmetrical end plate are Ci(ai) = Cb(ab,e) + (a2,e - at ) - (Ti,e , 0 < Hi < hi e ~ (ai,e < Cb(ab,e); i = pus) (61) To account for the difference in chord length distribution we make use of our weight functions wi belonging to the end plate, (i = p,s). In our optimization method each time we iteratively correct these weight functions such that the desired chord lengths (61) result. Table 4 Influence of the type of end plate planform for screws satisfying (60); screw ~ of Figure Lila) with two-sided shifted end plates; screw with one-sided end plates at suction sides of screw blades (k = ks; kp = O); screw with two-sided symmetrical end plates of Figure 13 (k = 2kp = 2kS). . screw w . ~ 6.10 . 8.20 6.45 8.20 6.10 8.20 Had/ see k 0.90 0.00 . ~0.70 0.00 . 0.55 0.00 AJA 0.789 0.815 0.802 0.815 0.780 - 0.815 Ehn 8.437 9.459 8.443 9.459 8.857 9.459 o6 see 1= 1.011 1.156 1.137 . 156 . 126 . 156 106Nm/ see ~ y(~)-~(.J 56.0 2.9 55.6 2.5 54.6 1.5 _ 96 % , end plate - two-sided, shifted one-sided two-sided, symmetric The results of Figure ll(a) and Tabled show some clear and understandable tendencies. Under the considered circumstances the best optimum propellers ordered with respect to decreasing efficiency ~ are as follows. First the propeller with the two-sided shifted t5] end plate, ~ of Figure llta), second the propeller with the one-sided end plate ~ of Tabled, third the propeller with the two-sided symmetrical end plate ~ Of t6] Table 4, and finally the propeller without end plates of Figure ll(a). That the application of a one-sided end plate appears to be somewhat less favourable than the application of a two-sided shifted end plate, was already predicted from some approximate considerations given in [33. To obtain some insight into the underlying propellers we have drawn in Figures 14-17 some results of the optimum propellers occurring in the Figurell(a) and Table 4, at the optimum values of ~ and k. In these figures we have chosen the leading and trailing edges of the screw blades symmetrically with respect to the 604 blade generator line, which is the line segment rh<~ OCR for page 585
DISCUSSION William B. Coney Bolt, Beranek and Newman, Inc., USA I wish to commend the author on his extension of classical optimization theory to include viscous losses. Several comments regarding the linearized approach are in order. End plates can effect propeller efficiency in two ways. The first, well described here, can be readily obtained under linear theory. The second effect a propeller forces comes from the axisymmetric, or near, loading on the end plates. It perhaps can be more easily seen in the aductor disk model for a dueled propeller in which efficiency gains arise which are associated with the percentage of the total thrust which is carried on the duct. Since the examples presented seem to be for fairly heavily loaded propellers, could the author comment on this? Also, can the theory presented here be extended to the wave adapted case in which the inflow is not uniform? AUTHORS' REPLY The first question concerns the fact that in the paper some results are presented for fairly heavily loaded screw propellers while the theory in the first instance is only a linearized one. As described in the paper there is included in the optimization model a more or less nonlinear effect, the so-called v-d-correction. However, this correction is only an approximate way to account for the complicated nonlinear character of the problem. From Figures 11 and 12 it is seen that although the nonlinear v-d-correction gives results which differ from the results of the linear theory; the same tendencies arise in the linear and nonlinear theories. Thus, it is hoped that the linearized optimization theory also for more heavily loaded propellers gives some information about the influence of end plates. The author agrees that thorough future research on nonlinear effects is important for the understanding of propellers with end plates of the type discussed, but he also believes that understanding of the phenomena observed by application of linearized theory is an essential first step. Regarding the second question as to whether the theory can be extended to the wake adapted case, I would like to refer to [9]. In that paper Klaren and Sparenberg incorporated in the linearized classical optimization theory for propellers with end plates, a method to deal with inhomogeneous inflow. It is possible to extend also the Optimization including viscosity discussed in the present paper to the case of inhomogeneous inflow by using their method. This can be done for the cases with and without the nonlinear v-d-correction. [9] Karen, L. and Sparenberg, J.A. "On Optimum Screw Propellers with End Plates, Inhomogeneous Inflow.. Journal of Ship Research, Vol. 25, No. 4., Dec. 1981, pp. 252-263. DISCUSSION Philippe Genoux Bassin d'Essais des Carenes, France How do your computations compare to experiments if there were any? AUTHORS' REPLY Some information about experimental work connected to the theoretical and numerical work carried out at the Department of Mathematics of the University of Groningen is given now. So far, experiments have been performed only on one propeller with end plates, namely the one designed by Sparenberg and de Vries, see [2]. It concerns a model screw propeller with end plates of the Two-sided symmetrical" type, an example of which is given in Figure 17. A disadvantage of that propeller model is that the theories used for optimization and for lifting surface design were linearized theories. This might be a reason that in the experiments the propeller model in the design conditions turned out not to deliver the thrust which was prescribed for the design. Another disadvantage is that the optimization theory used in the design was not as detailed as the optimization method described in the present paper. Furthermore, Table 4 illustrates that the used type of end plate planforms is not the most ideal for obtaining high efficiency. As explained in the present paper, application of Two-sided shifted end plate planforms is expected to give higher efficiency. Despite these drawbacks, the experiments with the model screw propeller revealed some promising features. It turned out that tip vortices could not be made visible, which is an indication that the basic principle of minimizing kinetic energy loss seems to work. Indeed, open water tests showed that the efficiency of the propeller with end plates for larger thrusts was significantly better than the efficiency of a corresponding B-screw propeller, even though it concerned model scale experiments, see [2]. Moreover the absence of tip vortices means that carefully designing propellers with end plates can probably help to reduce propeller noise. Application of types of end plate planforms that have relatively larger surface areas will presumably yield relatively higher noise reduction at the price of lower efficiency. To fulfill the need for further experiments with improved propeller designs with end plates, at the time of this writing this reply the geometry is calculated for a propeller model meant to be tested at MARIN in The Netherlands. In this new design we hope to diminish the shortcomings of the linearized theory by making use of the nonlinear v-d-correction in the optimization process as well as in the lifting surface design theory. Moreover, the end plates will be of the two-sided shifted type and the optimization process will be based upon the one described in the present paper. The author wishes to thank the discussers for their contributions. 605

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