Relation between the lifting surface theory and lifting line theory in the design of an optimum screw propeller

J M a r Sci Technol (2013) 18:145-165 D O I 10.1007/S00773-012-0200-3

O R I G I N A L A R T I C L E

Relation between the lifting surface theory and the lifting hne

theory in the design of an optimum screw propeller

K o i c h i K o y a m a

Received: 21 February 2012/Accepted: 30 September 2012/Published online; 23 December 2012 © J A S N A O E 2 0 1 2

Abstract A theory on an o p t i m u m screw propeller is described. The o p t i m u m means o p t i m u m efficiency o f a propeller, that is, m a x i m i z i n g thrust horse power f o r a given shaft horse power. The theory is based on the pro-peller l i f t i n g suiface theory. Circulation density ( l i f t den-sity) o f the blade is determined by the l i f t i n g suiface theory on a specified condition i n general. However, i t is shown that, i n the case of o p t i m u m condition, the circulation density is not determined by the l i f t i n g suiface theory, although the circulation distribution w h i c h is the chordwise integral o f the circulation density is detennined. The reason is that the governing equation o f the optimization b y the l i f t i n g suiface theory is reduced to that by the l i f t i n g line theory. This theoretical deduction is the main part of this paper. The importance o f the l i f t i n g line theory i n the design o f the o p t i m u m propeller is made clear. N u m e r i c a l calculations support the conclusion f r o m the deduction. This is shown i n the case o f f r e e l y running propellers and i n the case o f wake adapted propellers.

K e y w o r d s O p t i m u m screw propeller • Propeller l i f t i n g suiface theory • Propeller l i f t i n g line theory

1 Introduction

I t goes without saying that solving f o r the o p t i m u m pro-peller f r o m the efficiency point o f v i e w is the one o f the important of the w o r k to be done i n propeller design.

K . Koyama ( E l )

Retired, National M a r i t i m e Research Institute, 1-5-17 Iwadokita, Komae, Tokyo 2010004, Japan e-mail: koyaina_koichi@hotmail.com

Optimization o f the propeller operated i n the n o n - u n i f o r m flow field o f the wake behind a ship's h u l l is the one o f the important themes. The propeller f u l f i l l i n g the o p t i m u m condition is called the wake adapted propeller. V a n M a n e n [ 1 ] , Lerbs [ 2 ] , Hanaoka [3] studied the theme based on the propeller l i f t i n g line theory. Studies after t h e m were reviewed i n references [5, 6 ] . Recently a study on the optimization using the numerical l i f t i n g suiface theory was presented [ 7 ] . The author tried to obtain the o p t i m u m propeller based o n the propeller l i f t i n g surface theory [8, 9 ] . B u t the trial d i d not succeed and, on the contrary, the importance o f the U f t i n g line theory d i d became clear. I t became clear that the trial was impossible i n p r i n c i p l e . These are shown i n this paper. I n other words, this paper shows the importance o f the l i f t i n g line theory i n the design o f the o p t i m u m propeller i n the u n i f o m i i n f l o w and i n the n o n - u n i f o r m i n f l o w .

I n this paper the o p t i m u m propeller is considered based on the l i f t i n g surface theory w h i c h treats a vortex suiface system i n c l u d i n g l i f t i n g surface and t r a i l i n g vortex i n the potential flow. This p r o b l e m is to solve the c i r c u l a t i o n density of the bound vortex representing the blade f o r the given condition. Expressions o f the propeller l i f t i n g suiface theory based on helical coordinates are shown first. U s i n g the expressions, the energy equation o f the propeller is shown. O p t i m u m condition using the energy equation is transformed into the one of the calculus o f v a r i a t i o n . I t is shown theoretically that the calculus o f v a r i a t i o n based on the propeller l i f t i n g surface theory is reduced to the cal-culus o f variation based on the propeller l i f t i n g l i n e theory. A n d i t is shown h o w the theoretical result appears i n the numerical calculation. These are shown f o r the constant hydrodynamic p i t c h model f o r f r e e l y running propellers i n the first chapter and f o r the variable hydrodynamic p i t c h model f o r wake adapted propellers i n the next chapter.

2 A propeller in uniform inflow

Before going to the discussion on the o p t i m u m propeller i n n o n - u n i f o r m i n f l o w (propeller i n a h u l l wake) we consider the case o f the o p t i m u m propeller i n u n i f o r m i n f l o w (pro-peller i n open water) i n this chapter.

I n this case the model that the hydrodynamic pitch is constant radially is useful. The o p t i m u m propeller based on the model is discussed i n this chapter.

h = h' (1)

where h = h{r), h' = /?(/•'), and 2nh denotes the hydro-dynamic pitch. / denote radius. The model is used f o r the case o f the linear theory i n w h i c h the elements o f the free vortex sheet emanated f r o m the blade stay on the generated position due to negligible weak induced velocity. The model is also used f o r the case o f the non-linear theory based on the constant hydrodynamic p i t c h i n w h i c h the free vortex sheet is deformed by the induced velocity but the p i t c h o f the sheet is assumed to be constant radially.

I n this chapter we consider the case that a propeller advances w i t h a constant velocity V and rotates w i t h a constant angular speed Q i n still water. A x i a l and tangen-t i a l i n f l o w velocitangen-ty atangen-t tangen-the radius r on tangen-the propeller rotangen-tatangen-ting disc are denoted b y V, Or, respectively.

The bound vortex and the trailing vortex ai^e assumed to be on a helical surface. The pitch o f the hehcal surface is given by 2nh{r), where

h{r) V + Mr - ^ = t a n £ i = - , ^ — —

r Lir IV, (2)

and vt'a, vi^t denote axial and tangential induced velocity at the l i f t i n g l i n e representing the blade (Fig. 1). The pitch 2nh{r) is called hydrodynamic pitch. I t is assumed that the pitch doesn't vary i n the axial direction. I n general h{r) varies i n radial direction or /?(;•) is a f u n c t i o n o f r. B u t i n this chapter we assume that 2nh is constant i n radial direction using the average value o f h{r), w h i c h is shown by Eq. 1.

2.1 Propeller l i f t i n g surface theory using helical coordinates w i t h constant pitch

W e use the helical coordinates (T, a, /() w i t h the constant pitch 2nh which relates to the cyhndrical coordinates {x, r, 6) as fohows (Fig. 2)

x =

e

+ x/h, a = 6 - x/h, /t = r/h = 1 / tan EJ (3) T' = 0' + x7/7, a' = 9' - x'/h, p' = r'Jh = 1 / tan e[

(4) where the dash' denotes that these quantities are related to the point on the blade.

N o r m a l line to the helical surface is assumed approxi-mately to be the normal line to both the radius vector ;• and the h e l i x , then the n o m a l derivative is expressed by

8 8 . — = - c o s s i — + s i n f i i „ on ox rod 1

a

36 n 8 n 8 / t / 8T (5) The segment o f the helix of ji = const., a = const, is expressed by

d^ = + ffldT. ₍₆₎

Then the perturbation velocity potential f o r the flow around the propeller is expressed by

'•b ii y

where / denotes the number o f blades, integral means on the helical surface, r^, /'b denote propeller radius and boss radius, ^ i , S2 denotes the i' coordinates o f the leading edge and the trailing edge o f blade section, and y = y(s', / ) denotes circulation density on the blade, and

ds' = i f t V l + Ai'^dT'

(8)

J M a r Sci Technol (2013) 18:145-165 ₁₄₇ (9) f o T2 — = - COS 6; — + sm £i • dn" ' dx' ' = -Wl - Wn = J <^f' J yK [ ^ ~ ^ ' dT' f b n l \ 9 n 9 /(' y e r (10)

dr' / y ^ f l ^ ; / , , / / ) d . '

(18) A- - x f + r2 + ,-'2 _ 2n-' cos(0 - 0' - 2 r o « / 0 ' T — (7 T' — ff' 2 2 +(«2 + — 2 / i / i ' cos T + ff — T' — ff' 27i:7n / (11)

A s ff = ff' on the helical l i f t i n g surface representing blades, upwash on the blade w = w{s, r) is expressed by

8 ® 8 ^ J -^~ 471' l-l S2 OO

E / d'-' / rd.'

m=0 1 8«8«"7? d / . (12) Substituting Eqs. 5 and 8-11 into Eq. 12 we obtain

2 w • K . ^dr' I yK l - ^ ; / , , ; / ) d / (13) ]l^4i-'An 2 ^ 8^ 1 QnQn"R As" (14)

/i/t'+cos i / , 3 (^n/ - /(' sin v\„) {fi'v' - / i sin v,',,) \ ,

^3 ^5 (15) V = ( T - T ' ) / 2 , v' = ( r - T ' ) / 2 , v;„ = v ' - 2,n7tA dv' = hiT (16) ^ = \ / v ' 2 + + M'2 - 2/(^i' cos v;„

A s the upwash w consists o f the l i f t i n g line term Wi and the l i f t i n g surface teiTn >i'n, and kernel f u n c t i o n K{v; / i , / i ' ) has a singularity o f the 2nd order pole [ 3 ] , so w e get the f o l l o w i n g results.

tIa

-wi= J dfi' J yK{0-fi,fi')dT'

f b f l " ^ ^ ^ I yds'dr' y/lTja^ { r - r ' f (19) f o ^1 - v . u =

ƒ

dfl ^

yi^W

'o 2 1 • ; / ( , / ( d r d r ' / r ^ ( ° M ^ ; M , / / d / V l + ^('2 ( r - r ' ) ' where ^ ( v ; , , ; 0 = f ^ = ^ ^ ( v ; . , . ' ) {fl - fO {>• - r') = ^ ^ ( ^ ' ' ' ' ' ^ ' ' ) (20) (;? - ;7')^-(21) K^'Hv;fi,fi')=Kiv-fi,fi')-mfi,fi')-''^'^^'"^^''^''^ { f ' - f i ' Y (22) + cos v'

3{fiv' ~ fl'sin v'Jifi'v' - fl sin v'J R^

dv'

(23)

yds. (24)

I n the case o f the h f t i n g line theory, Wu (Eq. 20) becomes [3]

^ w n = ^ j y ^ d s '

2nJ s-s' (25)

This is deduced f r o m analyzing the singularity o f the integrand o f M'n.

Expressions shown above are f o r the theory o f constant pitch helical surface. I t should be noted that the radial component o f the flow is ignored as the effect is considered to be small i n this paper.

2.2 W o r k done by a propeller and energy loss o f a propeller

W o r k done against the suiTounding fluid by a propeller i n a unit time is expressed by

00)

rip 11 —{W* +u)ydrds (26)

where the integral domain is the l i f t i n g surface con-esponding to a blade o f the propeller, p is the density o f the fluid, W* + a denotes the velocity at the blade (Fig. 1)

W* = •\/{V + wf+{£>,•+ wf (27)

a denotes the velocity component due to the con-ection term

f o r the h f t i n g surface. p{W" + a)y implies l i f t density. 8<I>/9n is the upwash on the h f t i n g surface given by Eq. 12.

As shown i n Appendix 3, we have the relation

-{W* + Ü)

dn nr{V + + w,2} - V{Clr + w, +

wa}-Then the expression

(28)

P = 'pJJ y^'i"^ + i"a + w,2}drds

- lp JJy^i^'' + '''t + wajdrds (29)

is obtained. The torque and the thi'ust w o r k i n g on the small area di ds o f the l i f t i n g surface are denoted by dQ, dS,

dQ = lpyr{(W* + u) sin ei - vi'n cos £i}d/-ds

= lpyr{V + iva + wJidrds (30)

dS = lpy{{W* + ft) cos E] + Wu sin Sijdrds

= lpy{i2r + Wt + wajdrds. (31)

Then Eq. 29 becomes

P = a f[dQ-V [ f d S . (32)

The first term and the second term o f the right hand side of Eq. 29 i m p l y the shaft horse power and the thrust horse power, respectively, so P implies the energy loss o f the propeller. The above transformation shows that the w o r k done against sunounding fluid by the propeller (Eq. 26) is equal to the energy loss o f the propeller (Eq. 32).

Velocity component due to the con-ection term f o r the l i f t i n g surface ü is smaU, so i t w i l l be neglected below. P and P I (the h f t i n g line component o f P) and P n (the coiTection component o f P f o r l i f t i n g surface) are shown as f o l l o w s

P = P i + P n (33)

P=-lp / / yw*—drds

lo S2 lo J2

=lp I W*dr j yds _{^dr' / y'K{ ^—,p,p' Ids'}

lb Sl ''b a, (34) P I = -lp ffyW*~-drds = lp / W*dr / yds lp / t y * r d r d;-' / y'K{0-fi,p')ds' Y'mp,ii)dr' (35) B0tr = lp \ W*dr / yds .dr' lb S{ l-b Sl X jy'K^^)[—^,p^^nds' (36) 2.3 O p t i m u m propeller i n u n i f o r m i n f l o w

Our task is to seek the o p t i m u m solution so that the value P is m i n i m u m f o r the given shaft horse power. So P should be a m i n i m u m on condition that the first term o f P (Eq. 29) is specified. I n that case w e can get the solution by m a k i n g the f o l l o w i n g functional F m i n i m u m .

P = ^ p j j ky^riV + Wa + w^2)drds

J Mar Sci Technol (2013) 18:145-165 149

5F = 0 (38)

where k is the m u l t i p l i e r o f Lagrange. This is the f o r m u -lation f r o m the propeller l i f t i n g surface theory. The expression w i l l be transformed to the expression b y the l i f t i n g line theory below.

First o f all we have to investigate P i f U s i n g E q . 36 we can expand as f o l l o w s P i i = /p / W*dr f yds [ , ^ dr' S2 Sl Ki')r—L;p,,nds' '•b Sl

V

2 'o ^2 '•„ = lp j ( y + >i'a)dr ƒ yds j 2

V

1 +

dr'

Sl Si K(')i'--^;p,,nds'

lp{V + w.) I dr ƒ y(r,/Od. ƒ | y | ^ d .

rb Sl lb Sl X f p ' ) K ( ' ) ( l ^ ; p , Ads' (39) Here i t has been assumed

V + iVa = constant. (40)

Next, the relation f o r the kernel f u n c t i o n (cf. Appendix 2)

^ l ± i ; ^ . < « . , , ; . „ / ) . - / i ± i J . < » > ( - v . / . , . ) (41)

is used. When i n the final expression o f Eq. 39 f o r P n the order o f integration (drds) and ( d / d s ' ) is changed, and then the notations (/•, s) and ( / , s') are also changed f o r each o f the others, the expression P n becomes the same f o r m as before but changes its sign. So the value must be zero.

(42) Pn = 0

Then f r o m Eqs. 33 and 42

P = P i . (43)

These results shows M u n k ' s theorems [ 3 ] .

M u n k ' s theorem I I (Eq. 42): energy loss f r o m t w o b o u n d vortex elements vanishes f o r each other.

M u n k ' s theorem I (Eq. 43): energy loss is not altered when the bound vortex elements are displaced i n blade m o v i n g direction.

T h e situation is similar to the w i n g g o i n g straight on. H o w e v e r , the d i f f e r e n c e between the w i n g and the propeller blade should be noticed. I n case o f the w i n g , the m i n i m u m induced drag does not change w h e n one o f the l i f t element shifts i n the m o v i n g d i r e c t i o n according to M u n k ' s theorem I . I n case o f the propeller, i t is not thrust (or torque) but energy loss P is unchanged w h e n one o f the l i f t elements shifts i n the m o v i n g direction. I t is expected at this m o m e n t that the o p t i m u m h f t distribution not only radial but also i n a chordwise d i r e c t i o n is determined f o r a g i v e n torque. These results are the same as the results already shown b y Hanaoka [3, 4 ] .

N e x t w e have to investigate F (Eq. 37). The ratio o f the l i f t i n g suiface correction t e r m o f thrust to the l i f t i n g line term o f thrust is developed as f o l l o w s .

S j S i ^ l p y{wa)drds/lp J J ^ ( O r + Wi)drds

y{wn

sin£i)drds

/ //

yfi{V +

Wa)drds

(44) I n the meantime the ratio o f the l i f t i n g suiface c o n e c t i o n term o f torque to the l i f t i n g line term o f torque is developed as f o l l o w s .

Ö s / Ö i ^^pJJy(>"a2)drds J l p J J y i i V + Wa)drds

= JJyr{—wii cos ei)drds^ JJyr{V + Wa)drds

drds /{V + iva) / f yrdrds = 7'-\ Wn

y i j^'''^^I + y y ypdrds (45) Here we assume E q . 40 i n Eqs. 44, 45.

F r o m E q . 42

P n = -lpJJ yW*wudrds

= -lp{V + iva) J J y ^ l + fihv-adrds = 0. (46)

Then ' 1 +

y . wiiaras = 0. (47)

Using Eq. 47, i t is easy to show Eqs. 44 and 45 are the same.

Ss/Si = Q,/Qx = a (48)

Then E q . 37 can be developed as f o l l o w s .

F = lp{\ + ^-)JJkyQ>iV + w^)drds ~ lp{l + a)

j j yV{Q.r + wi)drds

lp{l + a.) / kfliiV + w^)[ / yds d / - - / p ( l + a

X / V{Q.r + Wt)[ / yds)dr

lp{l + a) / kTQ.r{V + vva)dr - lp{\ + a)

X / r y ( £ l r + M',)dr (49)

The o p t i m u m problem f o r the l i f t i n g suiface (Eq. 37) is transformed to the o p t i m u m problem f o r the l i f t i n g line (Eq. 49). The solution o f Eq. 38 is drawn not f r o m the l i f t i n g surface f o r m u l a t i o n but f r o m the l i f t i n g line formulation. So i t becomes clear that i t is impossible to determine the chordwise distribution o f l i f t f r o m the o p t i m u m energy condition.

The problem is solved by using Eqs. 38 and 49 and a specified shaft horse power

lp j j yiliiV + Wa -t- Wa2)drds = O, (50)

f o r the unknowns k and y. I t should be noticed that the condition (Eq. 50) is by the l i f t i n g surface theory.

2.4 N u m e r i c a l calculation

Our assignment is to seek the f u n c t i o n y w h i c h satisfies the m i n i m u m condition o f the f u n c t i o n a l F.

P= 'P j j y[kflr{V - COS Ei{wi + wn)}

-V{Q.r + sin £i (wi + i v n ) } ] d s d r

(51)

Using B i r n b a u m series f o r the mode f u n c t i o n , circulation density y o f the bound vortex is expressed as f o l l o w s

N=0

+ A P ) ( , ? ) e \ / w ^ + A P ) ( , , ) ^ 2 \ / r ^ , (52)

where <J denotes chordwise variable, i] denotes radial variable, X^iO denote B i r n b a u m series, and A^'^(;y) denote unknowns. Substitution o f Eq. 52 into E q . 51 gives

F = lp i y * ^ A W ( , ; ) ^ = 0

X J A A , ( ( ^ ) [ m 7 - { t / - c o s e i ( v i ' i + i v u ) }

X -V{Q.r + sin £i (wi + i v i i ) } ] d s d r . First variation o f Eq. 53 is

ÖF = lp / W*J2^A^"K'l) N=0 (53) •^2 X J / l A r ( ^ ) [ / : Q / - { t / - C O S £ i ( w i + W i i ) } -V{nr + s i n £ i ( w i + wn)}]dsd/- + ÖF2

SF2 = ip f f w*J2öM''\n)^^{^)

(54) d7-' / W*'^Ai'''\,f)X^,{a 2 N'=0 Ul s, X ( ^ Q ; - c o s £ i + V s i n E l ) ' ' l ^ 2 m i , , ' ) i i + j l 1 + f i ' - p,fi'] }ds' dsdr. (55)

I n the above expressions w j + vi'n is a f u n c t i o n o f y, so variation term f r o m vi'i + Wu gives 5F2. The detail is shown i n Appendix 5 as the Galerkin calculation method. Sometimes SF2 is neglected i n the calculation, i n w h i c h case we c a l l i t as being by M e t h o d I i n this paper. W e c a l l i t b y M e t h o d I I f o r the case w h i c h includes ÖF2.

U s i n g E q . 54 w e get the equation 5F = 0 i n w h i c h ÖA'^\I]) can be arbitrary, so the f o l l o w i n g expression is obtained.

J Mar Sci Technol (2013) 18:145-165 151

j l!,{^)[kQr{V^cosei(wi + Wu)}

Si

-V {Sir + sins I {wi + wu )}.ds

+ / MO n - l dr' W*'Y.A^''H,f)MO N'=0 X (/:Qrcosgi + V s i n e i ) ' •K{0;fi,n') l+fi'--;/(,// \ds' ds = 0 onN,r (56)

Equation 56 is tlie integral equations f o r y, as vi'i + vi'n is given by the integral o f the f u n c t i o n including y (Eqs. 19, 20). Equation 56 is the equation f o r M e t h o d I I . I n case o f M e t h o d I the second term o f the l e f t hand side o f the equation is neglected.

Equation 56 can be reaiTanged as f o l l o w s

Sl Sl + ƒ XN{0[kQr{-COS ey{yvi))-V{smEy{wi)}]ds Si + I i ^ ( 0 [ ^ : O r { - c o s £ i ( v i ' n ) } - V { s i n £ i ( v v i i ) } ] d s XN{0 n-\ dr' / w*'Y,A('''\n')MO N'=0 x{kQr cos Sl + y s i n e i ) ' 1 + / XNiO •K{0;fi,fi')\ds']ds

dr' / W/*'^A(^')(,/)A^-(^')

A"=0

X (kQr cos £i + y sin e i ) '

ll+Jl 1 + f f

T — T

•,fL,fi' ] }ds']ds = 0 onN,r.

(57) The second term o f the l e f t hand side o f Eq. 57 implies the l i f t i n g hne term o f M e t h o d I , the third term implies the l i f t i n g surface con-ection term o f M e t h o d I , the f o u r t h tei-m implies the l i f t i n g line term o f the addition f o r M e t h o d I I ,

and the h f t h term implies the l i f t i n g suiface conection term o f the addition f o r M e t h o d I I .

On the other hand, the first variation o f Eq. 51 is '•o S2

SF^lpJ j öy[kilr{V-cosEi{wi + Wii)}

rb Sl

— V{Qr + sin £i {wi + vi'ii)}jdsd;- + ÖF2

(58)

This is shown as the collocation calculation method i n A p p e n d i x 5. Using E q . 58 w e get the equation = 0 i n w h i c h öy can be arbitrary, so the f o l l o w i n g expressions are obtained i n case o f M e t h o d I .

kQ.r{V - cos £1 (vi'i + vi'ii)} - V{ilr + s i n £ 1 (vt'i -|- vt'n)}

= Oons,;- (59) I n case o f the l i f t i n g line theory vi'n = 0 and using E q . 2

the expression (Eq. 59) reduces to y { 0 ; - + v i ' i s i n £ i } _ V Q ; - { y - vi'iCosEi} Qh(r) Therefore, we get Q/c hir • const. (60) (61)

This is the linearized Betz condidon [ 6 ] .

N u m e r i c a l calculation was earned out by using Eq. 56 f o r t w o propellers. Propeller A is a conventional propeller and Propeller B is a highly skewed propeher. The blade contour is shown i n Fig. 3. The propeller is a f o u r bladed propeller. Particulars o f the propeller and the design condition are shown i n Table 1. The o p t i m u m propeller is solved o n condition that the advance coefficient / = 0.6456, and the torque coefficient = 0.0366.

Figures 4 and 5 show the results f o r propeller A i n the uni-foiTU i n f l o w . The chculation distiibution F is shown i n Fig. 4, and hydrodynamic pitch 2nh is shown i n Fig. 5. The value 2nh shown i n the figure is the value calculated by Eq. 19. The radial mean o f the value is used for the constant pitch.

F i g . 3 Projected contour of propeller A , B

Table 1 Particulars and design condition

Diameter D (m) 0.253 Boss ratio br 0.167 Expanded area ratio A^IAo 0.475 Number o f blades Z 4 Advance velocity y ( m / s ) 1.638 Number o f revolutions n (rps) 10.03 Advance ratio / 0.646 Torque coefficient 0.0366 C i r c u l a t i o n D i s t r i b u t i o n G= Tl'qiDV) ( P r o p e l l e r A) G 0 0.2 0.4 0.6 0.8 r/R

F i g . 4 Circulation distribution o f CP (propeller A )

The marks L S T and L L T show the value calculated by the l i f t i n g surface theory and the l i f t i n g line theory, respectively. The marks M I and M i l show the value by M e t h o d I and M e t h o d I I , respectively. The mark B 2 and B 4 show the value by the calculation using the 2nd term and the 4th term o f the B i r n b a u m series (Eq. 52), respectively. I f more than one term o f the B i r n b a u m series are selected, the equation cannot be solved, because the determinant is nearly zero.

A l l the results are almost the same. There is a small discrepancy between the results by M e t h o d I and the results by M e t h o d I I . Only the calculated hydrodynamic pitch b y M e t h o d I o f the l i f t i n g line theory shows a constant exactly. The results by M e t h o d I I o f the l i f t i n g surface theory coincide w i t h the results by M e t h o d I I o f the l i f t i n g hne theory, w h i c h verifies the theory shown i n Sect. 2.3.

Figures 6 and 7 show the results f o r propeller B i n the u n i f o r m i n f l o w . The results b y M e t h o d I I o f the l i f t i n g surface theory coincide w i t h the results b y M e t h o d I I o f the

Hydrodynamic P i t c h Inh ( P r o p e l l e r A) 0.45 0.4 0.3 —X—LLT,GaI,M I — 1 — L L T , G a i n I I — L S T , G a i n I,B2 —D—LST,GaU,l LB4 — © — L S I , G a i n II,B2 — e — L S T . G a l l ' l II,B4 —X—LLT,GaI,M I — 1 — L L T , G a i n I I — L S T , G a i n I,B2 —D—LST,GaU,l LB4 — © — L S I , G a i n II,B2 — e — L S T . G a l l ' l II,B4 \ < Y 0 0.2 0.4 0.6 0.8 1 r/J?

F i g . 5 Hydrodynamic pitch distribution of CP

C i r c u l a t i o n D i s t r i b u t i o n (7= r/(nD (<) ( P r o p e l l e r B)

T/J?

F i g . 6 Circulation distribution o f HS (propeller B )

l i f t i n g line theory also i n this case. The results by M e t h o d I o f the l i f t i n g surface theory don't coincide w i t h the others. The reason f o r the d i f f e r e n t performance o f M e t h o d I is explained as f o l l o w s . I n the case o f Propeller B the order o f the third term and the fifth term o f the l e f t hand side o f Eq. 57 becomes large, and the t w o terms are canceled b y each other i n case o f M e t h o d I I . So the contribution o f the third term i n M e t h o d I is the reason.

J M a r Sci Technol (2013) 18:145-165 153

Thrust coefficient Ki, torque coefficient K^^, and propel-ler efficiency = VS/(QQ) = JKJ{2nK,^ are shown i n Table 2. The results by the l i f t i n g surface theory 7i't(LST), /Cq(LST), ;?p(LST) are f r o m the integral (Eqs. 30, 31), whereas the results by the l i f t i n g l i n e theory ^^[(LLT), /^•^(LLT), )7p(LLT) are f r o m the integral (Eqs. 30, 31), where the l i f t i n g surface terms vi't2,Wa2 are excluded. Table 2 shows the comparison between the l i f t i n g hne calculation and the l i f t i n g surface calculation. I n case o f CP (propeller A ) , Ki, Kq b y the l i f t i n g surface calculation are smaller than those by the l i f t i n g line calculation by about 0.2 % . I n case of HS

0.45 Hydrodynamic P i t c h 2 jtA ( P r o p e l l e r B) 0.35 0. 3 0. 25 - L L T . G a U I I -LLT,Gal, 1,1 I I • L S T , G a l , l l I , B 2 - L S T , G a l , l l I , B 4 •LST,Gal,M II,B2 - L S T , G a l , M I I , B 4

(propeller B ) the former is smaller than the latter by about 1 % . The reduction rate is almost the same f o r Kt and Kq. A s a result, there is no discrepancy between i]^ o f propeher A and i]p o f propeller B .

3 A propeller in non-uniform inflow

I n the previous chapter we discussed the o p t i m u m propeller i n u n i f o r m i n f l o w . I n this chapter we w i l l discuss the o p t i m u m propeller i n n o n - u n i f o r m i n f l o w . So i n this chapter we assume that the hydrodynanuc p i t c h is not constant i n the radial direction.

/; 7^ h' (62)

This m o d e l is useful f o r the general case of a propeller, although the model o f constant hydrodynamic p i t c h used i n the previous chapter is useful f o r some special cases.

A propeller advances w i t h a constant v e l o c i t y Vs and rotates w i t h a constant angular speed Q i n a n o n - u n i f o r m flow field. A x i a l and tangential i n f l o w velocity at the radius /• o n the propeller rotating disc are denoted by Vi, Qir,

V, = Vs{l-w{r)}, O , r = 0 r { l - c o ( r ) } (63)

where w(r), co{r) are the axial and the tangential wake f r a c t i o n . W e ignore the circumferential variation o f the i n f l o w i n this paper. The special case o f w{r) = coir) — 0 coiTesponds to the case o f an open propeller i n the u n i f o r m i n f l o w . The pitch of the helical surface on w h i c h the bound vortex and the trailing vortex are located is assumed to be given by 2%h{r), where

Fig. 7 Hydrodynamic pitch distribution o f HS

hir) Vy + = tan fil =

r Ll\r + Wt

(64)

Table 2 Designed thrust, torque, and efficiency

i : t ( L L T ) 7(:t(LST) i r q ( L L T )

ATqCLST)

'/pCLLT) '/pCLST)

ratio ratio ratio ratio ratio ratio

L L T (Method 1) 0.2436 0.03660 0.6838 1.0005 1.0008 0.9997 L L T (Method I I ) 0.2434 1.0000 0.03657 1.0000 0.6840 1.0000 CP (Method I) 0.2439 1.0017 0.2434 0.9999 0.03666 1.0025 0.03659 1.0005 0.6835 0.9993 0.6836 0.9994 CP (Method n ) 0.2441 1.0025 0.2436 1.0005 0.03669 1.0033 0.03662 1.0014 0.6834 0.9992 0.6834 0.9992 HS (Method 1) 0.2419 0.9936 0.2401 0.9862 0.03684 1.0074 0.03657 1.0000 0.6746 0.9863 0.6745 0.9862 HS (Method H) 0.2456 1.0089 0.2430 0.9980 0.03700 1.0118 0.03660 1.0008 0.6820 0.9971 0.6820 0.9971

<Ö Springer

ds = ^h^l + f f i d i . (68)

F i g . 8 Inflow and induced velocity vector

and Wa, Wt denote axial and tangential induced velocity at the h f t i n g line representing the blade ( F i g . 8). T h e p i t c h 27i/j(/-) is called the hydrodynamic pitch. The p i t c h is a f u n c t i o n o f i n general. It is assumed that the p i t c h doesn't vary i n the axial direction.

Then the perturbation velocity potential f o r the flow around the propeller is expressed by

a

^

1

s'

where I denotes the number o f blades, integral means o n the hehcal surface, i^, denote propeller radius and boss radius, si, S2 denotes the s coordinates o f the leading edge and the trailing edge o f blade section, and y = y{s', ;•') denotes the circulation density on the blade, and

ds' = ^h'^/l+~f^dT'

ds" = h'^T+pl^dT'

(70)

(71)

3.1 Propeller l i f t i n g suiface theory using hehcal coordinates w i t h variable p i t c h

W e use the helical coordinates (T, cr, ji) w i t h the variable p i t c h 27i/7(;-) w h i c h relates to the c y l i n d r i c a l coordinates (A-, )•, 9) as f o l l o w s

dn'l

- cos e, + sm s, —

'

8A-'

' ft'Vl + m'2 1

^'7

8(7'

n 8

(72) P = A/(A- - A-'2) +

7-2

+ ,-'2 _ 2rr' c o s ( ö - 0' -

271777//)

T - f f , T ' - a ' \ , ,s2 / r. , , , , / ' l + ff-T'-ff' 27C777\ — /7 /7' + + / ' ' / » ' " 2/(/7/i'/7' cos

-2

2

7

\ 1 I ) (73)

T = 0 + A-//7, ( 7 = 0 - A-//7, /( = 7-//7 = 1 / taU (65) T' = 0' + A-'//7', (7' = 0' - A-'//?', / i ' = 7-'//7' = 1 / tan e',

(66) where the dash' denotes that these quantities are related to the p o i n t on the blade.

The n o r m a l hne to the helical suiface is assumed approximately to be the normal hne to both the radius vector r and the helix, then the n o i m a l derivative is expressed by 8 8 c - — = - cos £i — - I - sm 8i — 0771 GX 7-C / ï ^ T + T ? I V p) 90-

n

8

A 8

l l )

8T

Upwash on the blade w = ii'(j,7-) is expressed by

11' :

877]

1 / - I Sl OQ on blade ,„=0

8^

1

diiidn'lR

ds" lb Sl = J dr' J yh's/l+ii'2F^ds' f l 1 / - I

E

1

The segment o f the helix o f / j = const., cr = const, is expressed b y

877.1877'/P

(67) A n d using the notations

fdT' dT'. on blade (74) (75) (76)

J Mar Sci Technol (2013) 18:145-165 155 (77) we get 1

r ) }

l.i'h' sin I f , f-i + a - T ' - a ' 2mn\ ^ | w ^ + c o s ^ ^ . f z + ff-T' -a' 2nin\ \ 2 r j ' x ^ a , T ' - a ' , X < I -^r~h z h f x + a-T' -a' 2nm\ -/(/ïsm — I L J

wiiere f j , a' are given f r o m tlie value on the blade. Using the notations

= - + d . ' = - l d r ' 2 2 (78) (79) (80) V = — - — h h = —-—h —h 2 2 - uh' + u'h' = -h h + h' , , ^Ji-h' ^ x ^ — + { u - u ' ) — — -h'~{u-u') we get - ƒ / d r ' = ƒ / ( - 2 d M ' ) = ƒ - 2 / d « ' ;n=0 P3 1 W ' + C 0 S ^ „ - ^ ^ (81) (82) 27r7?z\ X < /t'v' - /(/j sin ( u' 2mn\ (83)

R = y v'2 + {fihf + (/i'/7'2) -2/;/2/('/7'cos (^i'' • (84)

Then upwash on the blade (Eq. 74) is expressed as f o l l o w s

w = Wl + Wn (85) Wi = 8 $ ; 8«i _onblade d r ' ƒ y ( s ' , r ' ) f t ' v ' ï + ^ F / d s ' r ( 7 - ' ) / 7 ' V l + / ' " r / d r ' (86) .33>ii 3 n i _onblade d r ' ƒ y{s',r')h'^l+ffiFl,ds' F> = f i ' + F i = ƒ 2/dw' — CO 0 Fl = J 2fdu' —OO It Fl, = j 2fdu' (87) 0 r ( r ) = ƒ r ( s , r ) d s . ^1 (89) (90) (91)

A n d vi'i coiTesponds to l i f t i n g line term (the contribution f r o m the free vortex o f l i f t i n g line theory) and Wn coiTe-sponds to the correction term f o r l i f t i n g surface (residual term w-wi f o r the l i f t i n g surface theory).

The expressions shown above are f o r the theory o f variable pitch helical suiface. I f we set h = h' i n the above expressions, the expressions are reduced to that f o r the constant pitch helical suiface w h i c h is shown i n Sect. 2 . 1 . The relation between the expressions i n Sect. 2.1 and i n Sect. 3.1 is shown i n Appendix 1. I t is to be noted that the definition o f the variable v' is different between Sect. 2 and i n Sect. 3. The radial component o f the flow is ignored as i n Sect. 2.

3.2 W o r k done b y a propeher and energy loss o f a propeller

W o r k done against the surrounding fluid b y a propeller i n a unit time is expressed by

•8<1>

dn {w; + ü)ydrds (92)

where the integral domain is the l i f t i n g suiface corresponding to a blade o f the propeller, p is the density of the fluid, Wi + ü denotes the velocity at the blade (Fig. 8)

W; = -^{Vi+wf + iQir + wf (93)

Ü denotes the velocity component due to the correction

term f o r the h f t i n g suiface. p{W*i + ü)y implies l i f t density. 8(D/8;;.i is the upwash on the l i f t i n g surface given by Eq. 74.

Similar to the case o f Sect. 2.2, we have the relation

8(D 0711

- y i { f i i r - | - v ( ' t + i V t 2 } . Then the expression

- lp J J yVi{D.ir + Wt + ivt2}d7-dj'

(94)

(95)

is obtained. The torque and the thrust w o i t e d on the small area d7-ds o f the l i f t i n g surface are denoted by dQ, dS,

dQ = lpyr{{Wl + u) sin 8i — wu cos ei }d7-dj' = lpyr{Vi + M'a - I - Ifa2}d7-ds

dS = lpy{(IVj* + i7) cos ei + wji sin ei }d7-ds = lpy{Q.ir + Wt + vt't2}d7-ds.

Then Eq. 95 becomes

O l d e VidS.

(96)

(97)

(98)

The first t e r m and the second term o f the right hand side o f Eq. 95 i m p l y the shaft horse power and the thrust horse power, respectively, so P implies the energy loss o f the propeller. The above transformation shows that the w o r k done against surrounding fluid by the propeller (Eq. 92) is equal to the energy loss o f the propeller (Eq. 98).

The velocity component due to the correction term f o r the l i f t i n g surface ü is small, so i t w i l l be neglected below. P and P I (the l i f t i n g line component o f P) and P n (the coiTection component o f P f o r the l i f t i n g surface) are shown as f o l l o w s P = P i + P n P = - l p 11 yW*~drds (99)

m

dill lo S2 lo S2 -lp J dr J Wlyds J dr' J wfy'ds' ll X / 2/di7' I'b S2 Vb St (100) dill lb S2

-lp I Ar J WlyAs ƒ Ar' J Wl'y'As'

lb Sl lb Sl

wr

IfAii' (101) 9 % 9771 lo S2 lo S2 Pii = - l p II yW^^drds ^ - I p J d r j W^yds J dr' J Wl'y'ds''^^^^^^ u X ƒ 2/d7(' (102) 0 3.3 O p t i m u m propeller i n n o n - u n i f o r m i n f l o w

Our task is to seek the o p t i m u m solution so that the value P is m i n i m u m f o r the given shaft horse power. So P should be a m i n i m u m on condition that the first term o f P (Eq. 95) is specified. I n that case, we can get the solution by m a k i n g the f o l l o w i n g f u n c t i o n a l F m i n i m u m .

F = lp JJkyilir{Vi - c o s £ i ( w i -|- i v n ) } d r d s

- lp JJyVi{Q.ir + sin£i(vi'i -|- ii'ii)}d7-d5'

<5P = 0

(103)

(104) where k is the m u l t i p l i e r o f Lagrange. This is the f o r m u -lation based on the propeller l i f t i n g surface theory. The expression w i l l be transformed to the expression b y the l i f t i n g line theory below.

Using the expressions F = kiü2 + n f ) - { v l + v',)

vl = lpJJ yVi(Oi7- + sin£i(ivi))d7-ds

= lp J J ' ^1 • ii'ud7'ds

Q.Q = lpJJyQi7-(yi - cosei(wi))d;-ds

= lp J J yü.ir{- cos Sl • wii)d;-ds

we get QÖ _ y S = _lpJJywlwidrds = Pi -lp J J yW^wiidrds = Pu. ^1 - v! (105) (106) (107) (108) (109) ( U O ) (111)

A t first the characteristics o f P n (Eq. 102) w i l l be investigated.

I n the expression P n the order o f integral Jj!j d7- J^J ds and d?-' J^^ ds' is changed, and then the notation (T, a, p) and (T', ( / , /(') are changed f o r each other.

J Mar Sci Technol (2013) 18:145-165 157 lo S2 lo F n = +lp j dr j W^yds j dr' S2 -11 ƒ Wfy'ds' J 2f'du' : i l 2 )

w^

(120) where l-l f.1 fl + cos u —

/?5 fi'v" — fill sin ( ll'

2mn

X <^ fiv - fi ll sm ( u —

•h'

2 2

Here the condition ( A )

T~a

„

h' + h

h-h' VR = [ll - U ) —— 0 is assumed f o r v' (Eq. 81), then

/ T - (T T' - a' , h + h' v ' = — ^ / ï — h ^ { u ~ u ) ~ ^ . (113) (114) (115) (116)

This assumption is also used i n the expression (Eq. 114). The sign o f v' changes when replacing (T, cr, h)-{x', a', h') f o r each other i n v ' and changing the sign of the integral variable u'. I n this change the f u n c t i o n ƒ (Eq. 83) does not change. Also, the sign o f m is also changed as the sign o f I n the above change the sign of u changes. As a result we have

11

2f'dii' = - ƒ 2/dH'. (117) 0 0

The detail is shown i n A p p e n d i x 4. Then we get

Pn = +lp / dr / W*yds

wt

S2 II

X ƒ W;'y'ds' J 2fdu'.

Jl 0

Next, i f the condition (B)

{in

w* w*

Q= Z = P = Ö ' ^ const. (119) h^/Y+lf h'y^ïTW

is assumed, the previous E q . 118 is reduced to

where

Q = W* _Vi+Wa _ilir + wt (121)

The expression Eq. 120 f o r P n is the same f o r m as the original expression Eq. 102 but changes its sign. So we get

Pn = 0. (122)

F r o m Eqs. 99 and 122

P = Pi. (123)

These results show M u n k ' s theorems [3] as i n Sect. 2.3. Equation 122 imphes M u n k ' s theorem I I . Equation 123 implies M u n k ' s theorem I .

F r o m Eqs. I l l and 122

^1 = Vl (124)

F r o m Eqs. 106, 108 and 64 the h f t i n g line terms are

Vl = l p j j y y i ( Q i r + s i n e i ( i v i ) ) d r d s

= lp f f y V i f i { V i - cos si{wi))drds (125)

(126) Q,2^lpJJ y Q i r ( y i - cosei(vi'i))drds

= lp [[yQ.ih{ü.ir + sinei{wi))drds.

Here, i f the condition (C) Vl _ Vl 2%r _ Vl Qir + w, QJI ~ ïhï-2nh " Cip- Vl + Wa is assumed, we get v l ^ k ' n l k' = const. (127) 128) U s i n g the above results, we obtain the final result. On condition ( A ) (B) ( C ) , Eq. 105 is transformed as f o l l o w s o f F = M Q ? + f ^ e ) - ( y , ^ + y | ) = m e ( i + ^ 1/5 = . ^ 2 f ( l « ) - K f ( l + i ) = ( l + ^ ) 'L l+a (129) where < ö Springer

K = k{l + a)/il+a/k').

(130) (131) This result means that the o p t i m u m p r o b l e m of the l i f t i n g surface (Eq. 105) is reduced to the o p t i m u m p r o b l e m o f the l i f t i n g line

(132) 1 + a/k' m l

I n other words, the solution o f Eq. 104 is obtained not by the l i f t i n g surface problem but b y the l i f t i n g line problem. This proves that the condition o f the m i n i m u m energy loss cannot determine the chordwise position o f the l i f t i n g element.

The above discussion is on the basis o f the assumption o f conditions ( A ) , ( B ) , (C). I n the special case of non-linear theory based on constant hydrodynamic p i t c h , the condi-tions ( A ) , ( B ) , (C) are realized exactly and so the above discussion is realized coirectly. This was already shown i n Sect. 2.3.

I n the condhion ( A ) (Eq. 115), \h - h'\ « 1 means near constant p i t c h , and \u — u'\ « 1 means short chord length (large aspect ratio o f blade). The condition ( B ) (Eq. 119) corresponds to the assumption (Eq. 40) used i n Eqs. 39, 44 and 45 i n Sect. 2.3. The condition (C) (Eq. 127) is f r o m the results o f Hanaoka's wake adapted propeller theory [ 3 ] .

The p r o b l e m is solved by using Eqs. 104 and 132 and a specified shaft horse power

lp yQ.ir{Vi - cos £i(wi + i v i i ) } d r d s = Qg (133)

f o r the unknowns k and y. I t should be noticed that the condition (Eq. 133) is by the l i f t i n g surface theory.

3.4 N u m e r i c a l calculation

Our assignment is to seek the f u n c t i o n y w h i c h satisfies the m i n i m u m condition o f the f u n c t i o n a l F (Eqs. 103, 163) (cf. A p p e n d i x 5). The f u n c t i o n a l F is also g i v e n b y another expression (Eq. 165) using the B i r n b a u m series (Eq. 164) f o r y. As shown i n Appendix 5, we have t w o numerical methods. The collocation calculation method is based on E q . 163 and the Galerkin calculation method is based on E q . 165. For each method we have two methods. M e t h o d I and M e t h o d I I . The 5F2 term is neglected i n M e t h o d I , whereas the «5^2 term is included i n M e t h o d I I . Integral equations o f M e t h o d I and M e t h o d I I o f the collocation method are given b y Eqs. 173 and 172, respectively. Integral equations o f M e t h o d I and M e t h o d I I o f the Galerkin method are given by Eqs. 182 and 181, respectively.

N u m e r i c a l calculation was c a n i e d out by the variable hydrodynamic pitch model f o r Propeller B i n n o n - u n i f o r m

i n f l o w o f the h u l l wake. Propeller B is a f o u r bladed h i g h l y skewed propeller. The blade contour is shown i n F i g . 3. Particulars o f the propeher and the design condition are

Table 3 Particulars and design condition

Diameter D ( m ) 0.253

Boss ratio br 0.167

Expanded area ratio AEMO 0.475

Number o f blades Z 4 Ship velocity Vs (m/s) 2.253 Number o f revolutions n (rps) 10.03 Torque coefficient 0.0366 1.5 = 0 = 0 = • = l-wt(r) 1 0.2 0.4 0.6 r/R 0.8

F i g . 9 Radial distribution o f i n f l o w velocity

0.06 0.05 0. 04 G 0.03 0.02 0 . 0 1

K

Ci-culation DistributionG^=//(TTJ^F) (Propeller B)

J M a r Sci Technol (2013) 18:145-165 ₁₅₉

shown i n Table 3. The radial distribution o f the wake is shown i n F i g . 9. Figures 10 and 11 show the resuhs.

Results shown are the calculation o f the equation

5F = 0 using the expression 177 or 167, where only one

term o f the Birnbaum series E q . 164 was selected i n gen-eral. I f more than one term o f the series are selected, the equation cannot be solved, because the determinant is nearly zero. This means that we can get the solution f o r the given chordwise distribution o f y, but that we cannot get the chordwise distribution o f y. This is the result f r o m the theory discussed i n Sect. 3.3.

A special case is the calculation method I o f collocation calculation method E q . 173. I n that case, the solution using more than one term o f the B i r n b a u m series is obtained. The solution y is considered to be one o f the indehnite solu-tions. F r o m the point o f v i e w o f the calculus o f variation, method I is not complete. Accurate numerical solution F should be the same as that o f the l i f t i n g line calculation. The deviation f r o m the l i f t i n g hne solution shows the inaccuracy o f the numerical solutions.

I n F i g . 10 the radial distribution of the circulation is shown. The radial distribution o f the hydrodynamic pitch is shown i n F i g . 11. A l l calculations i n Figs. 10 and 11 are a l i f t i n g suiface calculation using one term o f the B i r n b a u m series (Eq. 164). One group denoted by B 2 uses the 2nd term o f Eq. 164 and another group denoted by B 4 uses the 4th term o f Eq. 164. The m a r k M I and M i l show the value by Method I (Eqs. 173, 182) and M e t h o d I I (Eqs. 172, 181), respectively.

Hydrodynamic Pitch

2nli

(Propeller B)

The results by Method I I are almost the same as those by the l i f t i n g line theory w h i c h shows the h i g h accuracy o f M e t h o d I I . Calculation b y the l i f t i n g line theory, w h i c h is not shown i n Figs. 10 and 11, was performed by neglecting the l i f t i n g suiface correction term i n the expressions. The results by M e t h o d I are not the same as M e t h o d I I w h i c h shows an unsatisfactory approximation o f M e t h o d I .

4 Conclusions

W e have investigated the relation between the l i f t i n g sur-face theory and the l i f t i n g line theory i n the design o f the o p t i m u m propeller.

I n case o f the theoretical model w i t h constant hydro-dynamic pitch, w h i c h coiTesponds to the case o f a propeller i n open water, i t was shown that the governing equation o f the o p t i m i z a t i o n by the l i f t i n g surface theory is reduced to that b y the l i f t i n g line theory. So the integral equation w i t h more than one o f the terms o f the B i r n b a u m series does not give the solution. B u t w i t h one term o f B i r n b a u m series i t does. This means that the o p t i m u m circulation distribution w h i c h is the chordwise integral o f the c i r c u l a t i o n density can be determined but that the o p t i m u m circulation density can not be determined.

I n case o f the theoretical model w i t h variable hydro-dynamic pitch, w h i c h coiTesponds to the case o f a propeller i n a h u l l wake, the same conclusion is obtained provided that the three conditions are assumed.

The numerical calculation supports the conclusion. The conclusion is considered to be useful as the basis o f pro-peller design. — A — L S T . C o U I n,B2 — e — L S T . C o l M I I , B 4

ol

1 1 \ \ 0 0.2 0.4 0.6 0.8

1

r/X

F i g . 11 Hydrodynamic pitch distribution o f HS

A c k n o w l e d g m e n t s The author is deeply grateful to the late Dr. T. Hanaoka, whose w a r m guidance and encouragement have supported the author's l i f e o f study on propeller theory.

Appendix 1: Relation between the two expressions for the upwash

Upwash on the blade is expressed by Eqs. 18, 19 and 20 i n Sect. 2 . 1 . O n the other hand, i n Sect. 3.1 upwash o n the blade is expressed by Eqs. 85, 86 and 87. The expressions i n Sect. 2.1 are f o r the constant hydrodynamic p i t c h m o d e l and the expressions i n Sect. 3.1 are f o r variable h y d r o d y -namic p i t c h model. I f /; = /?' is set i n the expressions i n Sect. 3 . 1 , the expressions are reduced to the expressions i n Sect. 2 . 1 . So we obtain the relation between both expres-sions as f o l l o w s

(134)

h=h' -K \ii=i/

7I

n /!=/,'" / z 3 ( l + / t ' 2 ) " \ V 2 ft3(l+^i'2) 7f(0)| (135) (136) (137)

Appendix 2: Opposite symmetry of kernel function

W e w i l l prove E q . 41 using Eq. 23. W r i t i n g as

ƒ»(!'';/<,/(')

P3

3 ifiv' - fl' sin v;„) {fi'V - fl sin v'J P5 (138) i^(«)(v;M,M') = ^ ^ = = ƒ E / ; . ( v ' ; / v O d v ' . (139) Symmetry o f ft and /z' f,„{v';fi,fi') =f,„{v'-fi',fi) (140) (141)

is obtained. Next, opposite symmetry o f v w i l l be shown.

l-l

Y . U v ; f i , f l ) = Y . U - v ' - f i , f l ) (146)

is obtained. The function is the even f u n c t i o n o f v ' . So the kernel function w h i c h is an integral o f the f u n c t i o n , is an odd function o f v.

I&\v;fi,fl) = -K^'^\-v;n,fl) (147)

The combination o f Eqs. 141 and 147 can prove Eq. 4 1 .

Appendix 3: Proof of equation 28

Equation 28 is proved as f o l l o w s .

6<I>

-{W* + M) ^ = - ( W * + « ) ( w i + Wn)

= -{W* + ü)wi - W*wn - uwu (148)

The third term o f the final expression can be neglected due to smailness and f r o m F i g . 1 W* = Qr cos Sl +V sin si —Wl = Qr sin si — V cos e j . (149) (150) fm{v']H,fl!) flfl' + COS V

3 {fiv' - fl' sin v;„) {fi'v' - fl sin v,',,) Y/V'2 + fP- + i.i'2 - 2flfl' cos v;„ y/v'^ + fP- + fi'^ - 2fifi' cos v;,

^ + sinv' sin^

flfl' + (cos v' cos + sin v' sin^^)

y/v'2 + 1.1^ + fi'2 - 2flfl' (cos v' cos 2^p + siu v' siu 2^p)" (142)

3{/Jv'

— / i ' ( s i n

v'

cos — cos v' s i n 2 ^ p ) } { ^ V - / f ( s i n v ' cos 2nm cos V sm ' vin lim

f)}

^Jv'^ + fi^ + fi'^ - 2fifi' (cos v' cos + sin

v'

sin^f^)"

So

f,„{v'; fl, fl') = ƒ _ „ , ( - ! / ; A i , / f ' ) .

Then

/ „ ( v ' ; /(, fl') + A „ ( v ' ; /(, fi') =f,„i-v'; fi, fi')

+f-m{-v';fi,fi!). A n d when m = 0, U v ' - f i , f l ) = f i i { - v ' - f i , f l ) . Then, (143) (144) (145)

Then, we may manipulate Eq. 148 as f o l l o w s

- ( W * + i 7 ) — = ( t r + ( 7 ) ( J 2 r s i n £ i - V c o s s i )

— i V n ( n r c o s £ i + y s i n £ i )

= Q.r{{W* + u) sin fii — wn cos £1}

— y { ( t y * -|- ü) c o s f i i - l - v i ' n s i n e i }

= Q.r{V + iva + Wa2} " V{Q.r + + 1^2}

(151) where

J M a r Sci Technol (2013) 18:145-165 ₁₆₁

ll'a = — l l ' i COS £i, 11', = W'l Sin g]

11'a2 = 27 sin 6] — It'll COS Sl, it't2 = iVcosfii + It'll sin £i

• (152)

Appendix 4: Proof of equation 117

Equation 117 is proved as f o l l o w s . U s i n g the notation 1 1

ƒ„(«',

VÓ; ƒ ( , / / ) = V l + / ' V l + / ' ' ^ 1 , 2nm\

^ 3 , W ' + e o s ^ « - — j

- 3 ƒ , HI • [ I 2nm\ - ^ | / n ' - / . / i s i n ( ^ » - — j , 2nm X < V — sin U I I I h + h' I T - f f , T ' - G ' . h + h' ° 2 ., 2 2 we have f r o m Eqs. 83 and 113

l-l

=0 l-l

f ^ Y . U i i \ - v ' o ; f t ' , f i ) .

m=0

I t is easy to k n o w the relations. f,n(ii',v'Q; fl, fl')

=ƒ_,„(-«',

-i'ó;

fl', fl) f,„{ll, V'Q] fl, fl') +/-,„{"'> v'oi fl, fi')

= ƒ _ „ , ( - « ' , -i'ó; fl', fl) +f,„{-u', -vó; / i ' , fi) Mu', V'Q-, fl, fl') =M-u', -V'Q-, fl', fl) So l-l l-l Y^fmill',v'o; fl, fl') = J2fm{-ll',-V'o; fl', fl) m=0 Then, «1=0 l-l 2fdu'= / 2 ^ / „ ( M ' , v [ , ; ^ i , / 0 d « ' O '"=0 ; I-l = / 2

(-«',-v;,;

Ai',/Odi/'

m=0 l-l 2j22Mii",-v'o;fi',fi){-dii") 2f'du' (153) (154) (155) (156) (157) (158) (159) (160) (161) (162) Appendix 5: N u m e r i c a l method

Our assignment is to seek the f u n c t i o n y w h i c h satisfies the m i n i m u m c o n d i t i o n o f the f u n c t i o n a l F.

^^^P j j y[/<:f^i''{^i - c o s £ i ( v t ^ i + i v n ) }

- V l { Q i r + sin £i (vt'i +

vi'n)}]dsd;-(163)

Using the B i r n b a u m series f o r the mode f u n c t i o n , circulation density y o f the bound vortex is expressed as f o l l o w s

^ = 0

= A ^ ' K n y \ l ^ ^ + A ^ ' \ n ) A - e

+ A ^ ^ \ n ) ^ ^ f i A e + A ^ ' \ n ) e \ f ^ ^ (164)

where ^ denotes chordwise variable, if denotes radial variable, / I N ( 0 denote B i r n b a u m series, and A^^^iif) denote unknowns. Substitution o f Eq. 164 into Eq. 163 gives

F = ip [ W ; J 2 A ' - " H ' I ) S2 (165) X J XN{0[kQir{Vi - c o s S j ( w j + vt'n)} Sl - Vl { f i l + sin ei (vf I + vi'n)}]dsdr

Numerical calculation methods for ^7^ = 0 ai^e shown i n this Appendix. These expressions shown here are f o r a propeller i n non-uniform i n f l o w which is discussed i n Sect. 3. I f V i , /z' i n the expressions ai'e changed to V, il, h, the expressions become those f o r a propeller i n u n i f o i m inflow discussed i n Sect. 2. Here we show two numerical methods w h i c h ai'e by Eq. 163 and by Eq. 165. W e caU the calculation using Eq. 163 as being the cohocation calculation method, and we cah the calculation using Eq. 165 as the Galerkin calculation method i n this paper.

A t first the collocation calculation method based on Eq. 163 w i l l be shown.

A s vi'i and vfn are given by Eqs. 86 and 87, respectively, expression f o r F by Eq. 163 is ro S2 T = i p J J y[{k - mn-Vi 'b Sl - { m i r c o s f i i + Vl s i n e , } ƒ d r ' ƒ y'h'^l + fi'^ X < J 2fdu' + J 2fdii' >ds' dsdr (166) •Ö Springer

where ƒ is given by Eq. 83 of tlie main body. The first

variation of Eq. 166 is

ÖF = lp J J öy[mir{Vi - cos fij (ivi + ivn)}

rb Sl

-Vi{Qir + sinsiiwi +M>ii)}]dsdr + 5F2 (167)

ro J2

ÖF2 = lp J J ^[—{/rOircos El + Vl sinei}

rb Sl lb S2 X ƒ dr' ƒ 5y'h'\/T+^ lb Sl

0

ƒ 2fdu' + J l f d u ' i d s ' dsdr

(168)

The order of integral £° dr J^'^ ds and /^!° dr' J^J^' ds' is

changed for ÖF2.

lo S2 ÖF2 = ^P j f öy'h'^V+Jfi rb Sl lb S2

X J dr J {-y(/cQircossi + Vl sinei)}

-'•b Sl

X I ƒ 2fdu' + J 2/dH'|ds

ds'dr'

(169)

The notations (T, cr, p) and (T', a', p') are changed for each

other.

ro S2 ÖF2 = lp j J Sy i-b Sl r» S2 .1 f j

dr' I y'l/cQircosei

- f V i s i n f i J ' f t x / r + T ï ^

2f'du'+ j 2f'du')ds'

0

dsdr

(170)

where ƒ is given by Eq. 113 of the main body. As the result

we get

r» S2 5F = lp J ƒ öy[{k - l)QirVi i-b Sl iQ

- {/cQircosei + Vi sinei} ƒ dr' ƒ y / j ' ^ l + /('2

rb

S2

ƒ 2/d«' + ƒ 2 / d « ' l d 5 '

-00

0

J

dr' J 7{/cf2ircosei + Vi s i n e i } ' ^ ^ 1 + P^

1 St 0 2f'du' + J 2f'du'}ds'

0

dsdr

(171)

Then we get the equation öF = 0 in which öy can be

arbitrary, so the following expressions are obtained.

0 = ( / t - l ) Q i r y i

ro S2

- {feQir COS ei + Vl sinei} ƒ dr' J yh'^/T+Jfi

U 11 \

J 2fdu' + J 2/dM'Us'

dr' /

y{/cOircos81+Vi

sinei}'/?Vr+7?

rb Sl

0

. ; 2f'dii'+ J 2f'du'}ds' on s, r (172)

Equation 172 is the integral equation for y which we call

Method I I . I f the SF2 term is neglected in the calculation

for Eq. 172

0={k~l)nirVi - {kQ.1 r

cos ei + Vi sin

£1}

ƒ

dr'

j yU \/1

+ / i ' ^

Pb s\

0 " 1

j 2/d«' + j 2f dit'>ds' o n s , r (173)

-co

0

This Eq. 173 is the integral equation for y which we call

Method I . The final equation for the numerical calculation

is given by substitution of the Birnbaum series Eq. 164 into

Eq. 172 or Eq. 173.

J Mar Sci Teclinol (2013) 18:145-165

₁₆₃

0 = {k-l)QirVi

Co Jj

J J ^,^0

rb Jl

X {/cOjrcosEi + Vl sm El} h'y^T+Jif^

0 II ^

ƒ 2fdii' + J IfdiAds'

0

,1-1

- / d r ' / v v r ' x : ^ ' ^ ' ' ( ' / ) ^ * ( ^ ' )

J J ^/^o

I'b J]

X {^Qircosei + Vi sin£i}'/7.\/l + f f l

2f'du' + j 2f'du\ds' on N,r

0

0 = (k - l)airVi

n - l

(174)

dr' \r;Y.A^''\n')iAO

rb J,

X {^Qi;-cos£i + Vl s i n s i j / z ' v ' l + /('^

0 11

X

2fdu' + j 2fdu'\ds' on N,r (175)

Next, the Galerkin calculation method, which is based

on Eq. 165 using the Birnbaum series will be shown below.

From Eq. 165

11-1

F = ip / w;^A<-^H'i)Mmk-mirVi

N=0

lb

Jl

-{kQircosei + Vi sinei}

ro J2

X ƒ dr'l Wf'j^A^'^'K'fnN'iOh'VÏ+J'^'

rb Jl

U II

ƒ 2fdu' + J 2fdu' ids'

dsdr

(176)

First variation of Eq. 176 is

8F = lp f f Wl £ MW(7?)l^(a[(fc - l)QirVi

I-b

Jl

-{kQircos El + Vl sinei}(wi + vi'n)]dsd7- -|- ÖF2

(177)

ÖF2 = lp [ fwlJ2A^''\,i)M0[-{kQircosEi+Visinsi}

J 7

I'b Jl

rn J? ,1-1

dr' / < ; ^ M ( ^ ' ) ( 7 / ) A , I I ( ^ ' ) / 7 ' ^ / Ï T A 7 ^

I'b Jl

0

A''=0

dsdr

(178)

2/dH' + ƒ 2 / d w | d y

By changing the order of integral J'°dr£'-ds and

f'^°dr'J^^ds', and the notation (T, a, fi) and (T', a', fi!) are

changed for each other, we get

5F2 = lp [ f wiJ2^A^''K'i)M0

I-b s,

fdr' f Wl'J2A(''\,f');Ma

Ub Sl

{—(kQ.ircosEi + Vl s i n E i ) ' } h \ / 1 +

fP-dsdr

ƒ 2 / ' d i / + ƒ 2 / d M ' U 5 '

-00 0 j .

As the result, we get

n - l

ÖF = lp [ [ Wl'j2öA^''\>l)U0[{k-mirVi

i J,

ro J2

- / dr' fwl'Y^A(^'\,/)X,i{a

I-b

Jl

X {kClircosEi + Vl sinEijh'y/l +

fiP-0 " ~1

ƒ 2/d»' + ƒ 2/'d!/'id/

-00 0 j

; ,1-1

dr' / V F * ' ^ A ( ^ ' ) ( , / ) A ^ i ( ^ ' )

Ai'=0

I'b Jl

{fefiircosei + Vl s i n e J ' f t V 1 + P^

0 !, ^

-ƒ 2 / d i i ' + -ƒ 2/'d«' ids'

(179)

didr

(180)

Then we get the equation öF — 0 in which

SA''^\I'I)

can be

arbitrary, so the following expression is obtained.

0 = / A^ioiik - mi'-vi n-l X { / t Q i r c o s e i + Vi s i n s i j / z ' V l + ji'^ J 2fdu' + J 2fdiAds'

-oo 0

J - I dr' J w f ' j 2 A ^ ^ \ > l ' ) h i i ' ) N'=0

X {/<Qi;-cos8i + Vl sin Si}'h\/T+iJ?

0

-11

ƒ 2 / d « ' + ƒ 2 / d K | d / ds on A'', ;•

This is the integral equation f o r A^^\if) or y w h i c h we call M e t h o d I I . I f the ÖF2 term is neglected i n the calculation f o r E q . 181 •52 0 = ƒ A M i m k - m w i Sl J J ^ 0 . 182 'b Jl ^ ^ X { f c Q i j - c o s e i + Vl s i n £ i } / ! ' \ / l + /t''

0

II

X •

ƒ 2 / d i / ' + ƒ 2 / d M | d / d j on N, r

This E q . 182 is the integral equation for A'-'^\I] ) or y w h i c h we call M e t h o d I . The difference between Eqs. 174 and

175 i n the collocation calculation method and Eqs. 181 and 182 i n the Galerkin calculation method is the effect o f the chordwise integral f^^ Xff{^)ds.

Here w e have shown two numerical methods, the c o l -location calculation method based on E q . 163 and the Galerkin calculation method based o n Eq. 165. F o r each method we have t w o methods. M e t h o d I and M e t h o d I I . I n the above expressions Wi + wu is a f u n c t i o n o f y, so var-iation term f r o m vi'i -|- Wu gives ÖF2- Sometimes ÖF2 is neglected i n the calculation, i n w h i c h case w e call i t b y Method I i n this paper. W e call i t by M e t h o d I I f o r the case w h i c h includes

5^2-Appendix 6: Singularity of the kernel function

Kernel functions used i n Sect. 2.1 o f the main body are

K(v: 1.1,1.1') {Eqs. 15,21) mdK^°\v; fl, fl') {Eqs. 22,23). F r o m

the Ref. [3] the singularity o f . ^ ( v ; fi, fi!) is given as f o l l o w s

K{v;fi,ti')^{

in

l o , v < 0 (183)

As

K^°\v; fl, fl') = K{v; fi, fi') - K{0; fi, fi'). (184)

So K^'Kv;fi,fi')^. 0, v = 0 F r o m Appendix 1 / j 3 ( l + / i ' 2 ) 2 • / 7 3 ( l + / l ' 2 ) 2 K{v;fi,fi') K{0;fi,fi') (185) (186) (187) (188)

Appendix 7: Numerical method of singular integral

(Multhopp matrix)

Upwash o n the blade w is given b y Eq. 7 4 o f the m a i n body. Function F'' in the expression has the singularity o f the 2nd order pole at r = / as shown i n A p p e n d i x 6. The numerical integral can be done using the M u l t h o p p matrix as f o l l o w s [ 3 ] . [ r - r i - ( - i y ~ " ,-dr' dn' {'i-n'Y sin (pj

(cos.?,,-cos,p,.)"

jn+L. = j v ^ 7

(189)

(190)

4sinipj,

J Mar Sci Technol (2013) 18:145-165 165

A c c o r d i n g to the definition o f /t, j.i'

g = {hil - h'fi'f ƒ yh'^ï+^F^ds'

•Sl •52 = {Mh/h'-fi'f I yl^^^/ÏVW^F'ds'. R e f e n i n g to A p p e n d i x 1 h m g = l i m ( / f - / ( ' f / y h ^ ^ / ï + J / ^ F ^ d s ' S2 ; i 9 i ) (192) = limh^{l.i-l.i'f y-S2 ^ = l i " l ~ 7 T = 7 T / y^i—^'l^'P']<is' ' • - > ' • ' V T + ] ? 2 ; V 2 J Sl ds' (193) References

1. Van Manen JD, Troost L (1952) The design of ship screws of optimum diameter for an unequal velocity field. Trans SNAME 60 2. Lerbs HW (1952) Moderately loaded propellers with a finite

number of blades and an arbitrary distribution of circulation. Trans SNAME 60

3. Hanaoka T (1968) Fundamental theory of a screw propeller (especially on Munk's theorem and lifting-line theory) (in Japanese). Report of Ship Research Institute, vol 5, no. 6 4. Hanaoka T (1971) Fundamental theory of a screw propeller—^11

(non-linear theory based on constant hydrodynamic pitch) (in Japanese). Report of Ship Research Institute, vol 8, no. 1 5. 18th ITTC (1987) Report of the Propulsor Committee, load

optimization

6. BresUn JP, Andersen P (1996) Hydrodynamics of ship propellers, series 3. Ocean Technology, Cambridge, p 233

7. Lee C-S et al (2006) Propeller steady performance optimization based on discrete vortex method. In: 26th symposium on naval hydrodynamics, Rome

8. Koyama K (2010) On optimum propeller by the lifting suiface theory (in Japanese). In: Proceedings of the Japan Society of Naval Architects and Ocean Engineers, vol 10

9. Koyama K (2011) On the theoretical foundation of the lifting line theory for an optimum screw propeller (in Japanese). In: Proceedings of the Japan Society of Naval Architects and Ocean Engineers, vol 13