hydrodynamic characteristics of propeller blades in ... · 2 hydrodynamic characteristics of...

2 Hydrodynamic characteristics of propeller blades in curvilinear motion

Practical application of such methods is of some interest for a straight line motion ofthe wing. But for calculating dynamic characteristics of ship propellers much morecomplex vortex systems are needed [1,3,6].Vertical axis propellers have a circular disk which is flush mounted on the horizontalbottom of the ship and rotates about its central vertical axis. On the periphery of thedisk five or six vertical blades are arranged with the possibility to rotate about theirattachment axes according to definite low which secures the propeller thrust in anynecessary direction.Vertical axis propellers proved very effective on tugs and ferries, and are also used atthe ship’s bow to secure manoeuvering in crowded and restricted waters [7].The first vertical axis propeller, of Kirsten-Boeing type, was built in 1870. Its bladesmade a half revolution about their axis for each revolution of the whole propeller.A more perfected type of this propulsion, the Voith-Schneider propeller, wasintroduced in 1931. Its blades describe a complete revolution about their attachmentaxes for each revolution of the propeller disk.Beginning from 1960, some serious papers containing hydrodynamic analyses ofvertical axis propellers on the base of vortex theory, were published [2,3,6,10].However, most of the authors solved two-dimensional problems; moreover, they didnot take into account the great curvature of blades trajectories, or valued the influenceof finite aspect ratio of wings and curvature of flow running on the blades not enoughexactly. Therefore, often the design results considerably differ from real dynamiccharacteristics of vertical axis propellers [2].

2. Vortex system and calculation of inductive velocities

Let, the vertical axis propeller moves ahead with a constant speed Vp in idealincompressible fluid. The horizontal disk of the propeller rotates about its central axiswith a constant angular velocity w. There are z vertical blades arranged on a distanceRp from the disk central axis.The root section of the blade is very close to the disk, therefore, according to “mirrorimage” [7], the length of a wing’s bound vortex L is taken twice of the blade’s lengthLp. As a rule, Lp = 1.2 Rp and the average breadth of the blade is bp = 0.25 Lp.Circulation of the bound vortex is G(f) = Gm f(f), where

Gm - is the maximum value of the circulation;

f - is an angle determining location of the blade’s attachment axis on its orbit having

diameter D = 2Rp.Horizontal section of the propeller, by inverse of fluid motions, is shown in Figure 1,where:u = w Rp - circumferential velocity of the blade’s attachment axis;W - inductive velocity;

Zelik Segal and Alexander Segal 3

V - resultant velocity of fluid running on the propeller blade;Px and Py - respectively, longitudinal and transverse components of the lift force P.

Figure 1. Horizontal section of vertical axis propeller.

The components Px of all blades form the thrust of the propeller and must be directedahead in any place on the orbit. Therefore, the blade circulation G(f) in the first

(forward) semi-circle (0 £ f £ p) is directed against the angular velocity w , and in

the second semi-circle (p < f < 2 p) the circulation G(f ) has the same direction as

w ( Figure 1). In order to the thrust will have approximately equal values in the 1-st

and 2-nd semi-circles, it is necessary that the function of circulation f(f) would be an

odd periodical function :

f(f) = –f(2p – f) (1)

Theoretically the most optimal function is f(f) = ± 1, which changes its sign in points

f = 0 and f = p . This function secures uniform distribution of longitudinal inductive

velocities in jet far after propeller [11]. But practically it is impossible to have ajump-in change of the blade’s angle of attack in transition points from one semi-circleto another. Therefore it is useful to have a function of type f(f) = s in f , which

satisfies the condition (1) and secures enough rapid change of angles of attack intransition points f = 0 and f = p .


B y displacement of a blade from location f in location f + d f , circulation of thebound vortex changes on value dG. Then, according to Kelvin’s theorem, a freevertical vortex with circulation - dG is formed in this place.In the first approximation, the propeller is supposed a light loaded, i.e. the inductivevelocities are very small, that is why the free vortexes remain immovable, or – byinversion of fluid motions – they moves with a velocity of running on flow Vp .The horizontal (trailing or tip) free vortexes are formed according to Helmholtz‘stheorem [8, 11]. Consequently, the vortex system of a vertical blade is a set ofrectangular closed frames (Figure 2). Since G(f) is a continuous function, the framesare continuous vortex sheets moving in the propeller jet.The vortex sheets have a cycloid form according to trajectory of the blade axis(Figure 3):

x1 = Vp t + Rp sin f = Rp (f l + sin f)

y1 = Rp (1 – cos f) , (2)

where t = f / w - time;l = Vp /w Rp - advance coefficient of the propeller;x1 and y1 - respectively, longitudinal and cross axes having a beginning in immovablecentral axis of the propeller (by inverse motion).

Figure 2. Vortex system of a vertical blade.


Figure 3. Cycloidal vortex sheets.

Two vertical longitudinal planes, corresponding to f and f + df , are taken inFigure 3.Free vertical vortexes with circulation - dG , disposed between the planes, may besubstituted by a continuous vortex layer with circulation:

q = - dG(f) / s = -zGm df(f)/p Dl , (3)

where s = 2p Vp/z w - distance between free vortexes forming in one semi-circle ofthe blade’s orbit. The sign minus in expression (3) can be dropped by taking realdirection of free vortexes (Figure 3).Elements of horizontal vortexes having length dl and circulation G(f) , locatedbetween above-mentioned two planes, are equivalent to two infinite chains oflongitudinal and cross vortex elements with the same circulation G(f) - (Figure 4).The chain of longitudinal vortex elements with length dx 1 = Rp(l +cos f)df(Figure 4.b) is approximately substituted by a continuous vortex cord with circulation

G G Gc

m

sdx

z f d= =

+( ) ( )( cos )f f l f fpl1 2

(4)

The chain of transverse vortex elements with length dy1 = Rpsin f df (Figure 4.c) isapproximately substituted by a continuous layer with circulation per length

q s z f Dm1 = ( ) / = ( ) / G Gf f p l (5)


Figure 4. Elements of horizontal trailing vortexes.

Figure 5. Velocities induced by vertical free vortexes.


An arbitrary point A is taken on the blade’s attachment axis determined by an anglef0. Distance from the point A to upper and lower ends of the bound vortex are,respectively, L1 and L2 (Figure 5). The point A is a beginning of axes x and y.Circulation of vertical element with sides dx and dy, according to (3), is:

d qdxz

Ddf dxv

mG G= =

p lf( ) (6)

Velocity induced in point A by the vertical vortex element is determined by knownexpression based on Biot-Savart formula [8]:

dWd

x y

v=+

+G

4 2 2 1 2p

q q(cos cos ) (7)

where angles q1 and q2 are shown in the Figure 5.Vector of velocity dW is perpendicular to the plane containing the vortex element andpoint A. Therefore, longitudinal and transverse projections of the velocity dW areequal:

dW dWy

x yx =

+2 2; dW dW

x

x yy =

+2 2 (8)

Integrating the expressions (8) on the whole volume of propeller jet and taking in

account formula (7) and designations C = z Gm/4p2Dl; r x y L12 2

12= + + ;

r x y L22 2

22= + + , the longitudinal and transverse velocities in the point A, induced

by free vertical vortexes, will be:

W Cy

x y

L

r

L

rdf dxx

b

=+

+•

ÚÚ 2 20

21

1

2

2

pf( ) ( ) (9)

W Cx

x y

L

r

L

rdf dxy

b

= -+

+•

ÚÚ 2 20

21

1

2

2

pf( ) ( ) (10)

where b = Rp (sin f0 – sin f); y = Rp (cos f – cos f0) (11)

An area on the upper surface of propeller jet with sides dx and dy, according to Figure4, has a longitudinal vortex element with circulation Gc and cross vortex element with

length dy = Rp sin f df and circulation G1 = q1dx (Figure 6).


Figure 6. Velocities induced by horizontal free vortexes.

The longitudinal velocity induced by cross vortex element in the point A, accordingto Biot –Savart formula [1, 8], is:

dWdy

rx1

1

124

=G sin cosu j

p (12)

where u - angle between the vortex element and radius-vector r1 connecting the

vortex element with the point A (Figure 6); sinu =n

r1

1; cosj =

L

n1

1; sin cosu j =

L

r1

1.

Similar formulas with values r2 and L2 correspond to lower surface of the propellerjet.The longitudinal element of upper vortex surface with length dx and circulation Gc

induces a transverse velocity in the point A (Figure 6). Projection on axis y of thisvelocity is equal:

dWdx

ry

c1

124

= -G sin cosb V

p(13)


where sin b = l1/r1; cos x = L1/l1; sin b cosx = L1/r1 .Integrating the expressions of type (12) and (13) about the upper and lower surfacesof the propeller jet gives the longitudinal and transverse velocities in the point A,induced by free horizontal vortexes:

W CRL

r

L

rf d dxx p

b1

1

13

0

22

23= +

•

ÚÚ ( ) ( )sinp

f f f (14)

W CRL

r

L

rf d dxy p

b1

1

13

0

22

23= - + +

•

ÚÚ ( ) ( )( cos )p

f l f f (15)

Taking the integrals (9), (10), (14) and (15) on x (see [5] - table integrals No. 146,206 and 267) gives:

W CL

y

L

y

bL

yr

bL

yrdfx = + - -Ú (arctan arctan arctan arctan ) ( )1

0

22 1

1

2

2

pf (16)

W Cr L r L

r L r Ldfy = -

+ +- -Ú0 5 1 1 2 2

1 1 2 20

2

. ln( )( )( )( )

( )p

f (17)

W CRL

y L

b

r

L

y L

b

rf dx p1

12

12

0

2

1

22

22

21 1=

+- +

+-Ú [ ( ) ( )] ( )sin

pf f f (18)

W CRL

y L

b

r

L

y L

b

rf dy p1

12

12

0

2

1

22

22

21 1= -

+- +

+- +Ú [ ( ) ( )] ( )( cos )

pf l f f (19)

where r1 and r2 contain, instead of x, the lower limit of integrals b.Numerical integration of expressions (16) – (19) doesn’t present any difficulties andallows to determine longitudinal and transverse velocities induced by free vortexes inany point of orbital surface of the vertical axis propeller.

3. Two-dimensional problem

In some particular cases expressions (16) – (19) are integrated exactly. Now, it willbe considered a two-dimensional problem: L1 = L2 = • , consequently, W1x = W1y = 0and r1 – L1 ª 0.5 (b2 +y2)/L1.

Taking in account (1) and df ( )fp

=Ú 00

2

, equations (16) and (17) are reduced to:


W C signyb

ydfx = -Ú ( arctan ) ( )p f

p2

0

2

(20)

W C b y dfy = +Ú ln( ) ( )2

0

22

pf (21)

When the point A(f0) is in the first semi-circle ( 0 0£ £f p ) the function sign y = (+)

by integration anti-clockwise from 2p-f0 up to f 0, and equal to (-) by integration

from f0 to 2 p - f0 ( Figure 7.a). When the point A(f0) is in the second semi-circle

(Figure 7.b), the function sign y = (+) by integration anti-clockwise from f0 up to

2p - f0, and (-) from 2p - f0 up to f0 .

Figure 7. Design point A in 1-st or in 2-nd semi-circle.

Taking in account the condition (1), the first part of integral (20) is reduced to:

1. 0 0£ £f p : p f p f p f p ff

p f

p f

fpC signy df C df C df Cf* ( ) ( ) ( ) ( )= - =

-

-ÚÚÚ 4 0

2

20

2

0

0

0

0

(22)

2. p f p< <0 2 : p f p f p f p fp f

f

f

p fpC signy df C df C df Cf* ( ) ( ) ( ) ( )= - = -

-

-

ÚÚÚ 4 02

2

0

2

0

0

0

0

(23)


For the second part of integral (20) it is used follow transformations:

arctan arctansin sincos cos

cot(cot )b

yarc=

--

= -+

= -+f f

f fp f f p f f0

0

0 0

2 2 2 2

For main value of arccot it is: 02

0<+

<f f p , that gives: - < < -f f p f0 02 ,

consequently:

- = --

- = - - +Ú Ú ÚÚ-

-

-

-

-

-

2 22 2

0

20

2

0

22

0

0

0

0

0

0

Cb

ydf C df C df C dfarctan ( ) ( ) ( ) ( ) ( ) ( )

p

f

p f

f

p f

f

p f

f p f f f p f f f f

The first integral in the right part of the last expression is equal to zero since f(-f0) =

= f(2p - f0); and the second integral is taken by parts, using that integral of an odd

periodical function for a whole period is also equal to zero:

- = [ ] - = -Ú Ú--

-

-

2 20

22

2

00

0

0

0

Cb

ydf C f C f d Cfarctan ( ) ( ) ( ) ( )

p

fp f

f

p f

f f f f f p f (24)

Adding the expressions (22) and (23) with (24) and confining it by f(f0) =

sin f0 gives:

1. 0 0£ £f p : Wx = 2p C sin f0 (25)

2. p f p< <0 2 : Wx = –6p C sin f0 (26)

In the second semi-circle sinf0 < 0, therefore in all points of the orbit the longitudinalinductive velocities Wx are directed as the flow velocity Vp. And in the pointssymmetrical about cross diameter, longitudinal velocities in the second semi-circleare three times more than in the first semi-circle. For calculation of the transverseinductive velocities, the values b = Rp (sin f0 – sin f) and y = Rp (cos f – cos f0) aresubstituted in (21). Then, after simple transformations and taking in account thecondition (1) the expression (21) is reduced to:

W C dfy =-Ú22

0

0

2

lnsin ( )f f f

p(27)

Taking f(f) = sin f and integrating (27) by parts gives:


W C dy = --Ú sin cotf f f f

p0

0

2

2 (28)

Addition of integral --

=Ú sin cotf f f fp

00

20

20d to (28) and taking in account that

cotcos cossin sin

f f f ff f

-=

+-

0 0

02, allows to obtain a final formula for transverse velocity at

L = •:W Cy = -2 0p fcos (29)

4. Numerical analysis of inductive velocities

Numerical integration of expressions (16) – (19) was made for G G( ) sinf f= m ,l = 0.6 and Lp/D = 0.6 at various positions of propeller blades on the orbit and somepoints along the blade axes.Calculation results have shown that inductive velocities Wx, W1x, Wy and W1y – eachseparately are distributed along the blade not uniformly; but sums Wx + W1x andWy + W1y are constant along the blades with high exactness, except a very small part(2-3%) of the blade length, near the end of the blade.

Table 1 presents calculation results of inductive velocities, divided by Cz

Dm=

G4 2p l

,

for finite blade length and also for 2-dimensional problem according to formulas (25),(26) and (27). As seen from Table 1, difference between velocities of finite andinfinite blades is only 4 – 12 % for longitudinal velocities, and about 25 – 90 % fortransverse velocities. The less differences are in the second semi-circle of the orbit.It is worth also to calculate dimensional values of the inductive velocities, inparticular, to compare them with the velocity of running on flow Vp.

Table 1. Dimensionless inductive velocities at G(f) = Gmsin f.

f, deg. 0 30 60 90 120 150 180 210 240 270 300 330

Wx+W1x 0.00 2.81 4.86 5.61 4.86 2.81 0.00 9.82 17.00 19.64 17.00 9.82

Wx (2-dim.) 0.00 3.14 5.44 6.28 5.44 3.14 0.00 9.42 16.32 18.84 16.32 9.42

Wy+W1y -10.70 -9.46 -6.02 -1.32 3.27 6.81 8.06 6.81 3.27 -1.32 -6.02 -9.46

Wy (2-dim.) -6.28 -5.44 -3.14 0.00 3.14 5.44 6.28 5.44 3.14 0.00 -3.14 -5.44

The lift force of a bound vortex is determined according to formula of Joukowsky[8]:


P VLp= r fG ( ) (30)

where r = 1000 kg/m3 r is the water density.Since the force P is directed perpendicular to resultant velocity V, the longitudinalcomponents of it: Vp, Wx and W1x do not create the propeller thrust, and the transverseinductive velocities are nearly symmetric about longitudinal diameter of thepropeller. Velocities induced by bound vortexes are relatively small. Consequently,for calculation of the propeller thrust T it can be used only the cross component ofrotary velocity of the blade usin f :

Tz uL

d zuLm p

m p= =Úr

pf r

pGG

20 52

0

2

sin . (31)

For example, it is taken a vertical axis propeller with follow parameters: D =2.0 m, Lp

= 1.2 m, bp = 0.3 m, a = 0.3, z = 6, n = 105 r.p.m, u = 0.5 D w = 11.0 m/s, Vp = 6.6 m/s,

l = Vp/u = 0.6, T = 40000 N, s = 2T/r D LpVp2 = 0.765. Then, according to (31), the

value of maximum circulation will be Gm = 1. 01 m2/s.

After determination of Cz

Dm=

G4 2p l

= 0.128 m/s, the dimensional values of all

inductive velocities can be calculated according to data of Table 1. For example, themaximum longitudinal velocities in 1st and 2nd semi-circles are, respectively,Wm1 = 5.61 ¥ 0.128 = 0.72 m/s, and Wm2 = 19.64 ¥ 0.128 = 2.51 m/s. These values are

enough great relatively to flow velocity Vp = 6.6 m/s.In conclusion of this part, it must be noted, that in formulas (4) - (6) the distance

between free vortexes sV

zp=

2pv

is taken on assumption that by motion of the

propeller with velocity Vp the free vortexes remain immovable (or by inversion ofmotion they move with the flow velocity Vp). But in reality they move owing topresence of inductive velocities. It can be approximately taken a conditional increaseof the velocity Vp by a some value Wc. If this value will be equal to a half sum ofaverage longitudinal inductive velocities on the first and second semi-circles of theorbit, then at s = 0.5 – 0.8 the design inductive velocities will be decreased on 10 - 15

%. But for a real function G(f) the most intensive free vortexes are placed near side

layers of the propeller jet where longitudinal inductive velocities have the least values(see Table 1). Moreover, in these layers cross inductive velocities cause on thevertical free vortexes lift forces directed against the velocity Vp.

All of these factors allow to consider the decrease of obtained inductive velocities ats = 0.5 – 0.8 not more than 7 – 10 %.


5. Experimental hydrodynamic characteristics of blades

If vectors of flow velocities at leading and trailing edges of a wing do not directed onone line, then the flow about the wing is curvilinear. Such a flow along the profileoccurs, in particular, when a longitudinal axis of a wing moves along a curvilineartrajectory.According to equations (2), relative radius of the trajectory’s curvature of a propellerblade R R b= / on dependence of angle f and advance coefficient l is presented inFigure 8 at b = 0.15 D [5]. As seen from Figure 8, on a sufficient part of the orbit R <4 – 5. For such small values of R hydrodynamic characteristics of wings in a realfluid, even by small angles of attack, essentially differ from analogous characteristicsin straight line flow. Therefore, experimental characteristics of wings in curvilinearmotion are necessary for design of vertical axis propeller. They are also a methodicalinterest in general theory of wings.The experimental determining of wing hydrodynamic characteristics was carried outin a circular testing basin of S.-Petersburg Water Transport Institute.Four models of wings with a chord b = 30 cm and length L = 40 cm were tested.Three of the models had a symmetrical profile NACA with different relativethickness d: 0.10, 0.15 and 0.20. The profile of the fourth model was derived fromNACA-0015 section shape by bending its axis on radius Rc = 6.0 b.The basin bottom was leveled in horizontal plane with accuracy about 1mm.Clearance between the lower butt-end of the model and the basin bottom wasapproximately 3 mm, and the upper part of the model pierced the free surface ofwater. At all relative radiuses of the circular motion of the wings from R = 1 to R =7, the velocity of the attaching axis O was equal to V = 0.45 m/s. The correspondingFroude number V/ gb = 0.26 secured a practically waveless mode of the wingmotion. Consequently, the basin bottom and the free surface of water can beconsidered as plane walls, allowing to assume infinite wing span [1, 8].Detailed description of this experiment and obtained results are given in the work [9].This paper presents only data necessary for calculation of lift forces and angles ofattack of propeller blades. An important parameter for hydrodynamic characteristicsof wings in curvilinear motion is a distance from attaching axis O to the leading edgeA. In the experimental data and calculations it is used a relative value a = AO/b. Theexperiment was carried out at a = 0.425, but all the results can be simple converted toany value of a [9].

It is known that derivative ∂∂Cy

a and angle of zero-lift direction a0 practically do not

depend on the Reynolds number [8, 11]. Therefore, values of a0 and ∂∂Cy

a obtained

in experiment at Re = 1.3¥105 are available too for the full scale conditions.

Moreover, the experimental data were compared with known results of wind tunneltests performed in straight line flow [1, 11]


For converting this results to infinite wing span, the known following formulas havebeen used [8, 11]:

( )∂∂

=+

C Bk

ky

ka 2;tana

piyC

k= , (32)

where

( )∂∂Cy

ka - derivative corresponding to tunnel tests of wings with a finite span k;

B - derivative corresponding to infinite span;ai - down wash angle between lift force and perpendicular to wing chord.According to theoretical investigations, B = 2p, but in real fluid for d = 0.10 – 0.15according to tunnel tests and experiments in curvilinear wing motion, it isapproximately A = 5.73. Therefore, when the effective angle of attack is in degree,then

Cy = 0.1 (a – ao) (33)

whereCy - lift coefficient for infinite wing span;a - angle of attack between profile chord and resultant velocity of water V.The maximum lift coefficient Cymax increases, as a rule, about 20 – 30 % by increaseof the Reynolds number from 105 to 107 [6, 7]. In any case, by design of vertical axispropellers the effective angles of attack, as a rule, does not exceed 8 – 10o, which aredisplaced in the straight line part of the curves Cy = f (a – ao).Angles of zero-lift direction a0 in dependence from relative curvature K = 1/R andrelative distance a are presented in Figure 9.Analysis of the hydrodynamic characteristics of the fourth model has shown that thebend of the profile center line is equivalent to a corresponding curvature of thewing’s motion. Therefore, to use experimental results, an equivalent curvature offlow must be calculated by following formula

Ke = 1/ R – 1/ R c (34).

6. Blade kinematics of vertical axis propeller

Rotation of propeller blade about its attachment axis is determined by angles gbetween profile chord and tangent line to the blade’s orbit. Dependence g = f(f) isnamed as blade kinematics.


In the first Foith-Schneider propellers a normal kinematics was used, when allperpendiculars to the blade’s chords have crossed in one point – regulation pole.Ratio of the pole eccentricity to orbit radius Rp is named as relative eccentricity eo.

Normal kinematics is expressed by dependence

g ff

=+

arctansin

/ cos1 eo(35)

According to formula (35) sinus of maximum angle g is equal to eo. Therefore, for

any dependence g = f( f) relative eccentricity is determined conditionally as

eom m=

+sin

( )g g1 2

2, where g m1 and g m2 are maximum angles, respectively, in the

first and second semi-circles of the blade’s orbit.In modern vertical axis propellers the blade kinematics is secured by a multi-linkmechanism, where each blade is connected by several links and hinges with oneregulating pole. The value eo in such kinematics achieves 0.82 – 0.84.In test models of vertical axis propellers, as a rule, a peripheral cam imparts anydesired motion of the blades. And there are projects to use cam mechanisms also inregular propellers.The angle g is calculated as a sum:

g b a= + (36)

where b – angle between velocity V and tangent line to the blade’s orbit.

The lift force of a blade is determined by a known formula of wing theory [1, 4, 8]:

P CV

L by p p=r 2

2(37)

Equating the right parts of (30) and (37) gives:

C Vby p= 2G ( ) /f (38)

Calculation of blade’s kinematics for above-mentioned example of vertical axispropeller is presented in Table 2. In first rows of the Table 2 there are longitudinaland transverse inductive velocities decreased (in comparison with data of Table 1) by10%, taking in account the motion of free vortexes. The angle q between the fluid

resultant velocity V and axis x is calculated in row 7. Then it is determined angles bbetween the velocity V and tangent line to blade’s orbit according to 12 values of theangles f. Lift coefficient of the blade Cy is calculated in row 10 according to formula

(38), taking in account G G( ) sinf f= m and Gm = 1.01 m2/s. The effective angle of


attack is calculated by formula (33) as for infinite wing span, since the inductivevelocities were defined by means of a three-dimensional vortex system. Relativeradiuses of flow curvature are obtained from Figure 8, and only in the point f = 180°

it is taken R = 1.0. Equivalent curvature of the flow Ke for blades with a bent profilecenter line is determined on expression (34). The angle of zero-lift direction a0 is

taken from Figure 9 by a = 0.3. The angles g are negative when the leading edge of

the blade is inside of the orbit.

Table 2. Calculation of blade kinematics at G(f) = Gm sin f.

1 f degree 0 30 60 90 120 150 180 210 240 270 300 330

2 Wx+W1x m/s 0.00 0.32 0.56 0.65 0.56 0.32 0.00 1.13 1.96 2.26 1.96 1.13

3 Wy+W1y m/s -1.23 -1.09 -0.69 -0.15 0.38 0.78 0.93 0.78 0.38 -0.15 -0.69 -1.09

4 Vx = Vp+Ucosf+Wx+W1x m/s 17.60 16.45 12.66 7.25 1.66 -2.60 -4.40 -1.80 3.06 8.86 14.05 17.26

5 Vy = Usinf+Wy +W1y m/s -1.23 4.41 8.83 10.85 9.90 6.29 0.93 -4.71 -9.15 -11.15 -10.22 -6.59

6 tan q =Vy / Vx -0.07 0.27 0.70 1.50 5.96 -2.42 -0.21 2.62 -2.99 -1.26 -0.73 -0.38

7 q = Atan q * 57.3 degree -4.0 15.0 34.9 56.3 80.5 -67.5 -11.9 69.1 -71.5 -51.5 -36.0 -20.9

8 b = f - q (-180 or -360) degree 4.0 15.0 25.1 33.7 39.5 37.5 11.9 -39.1 -48.5 -38.5 -24.0 -9.1

9 V = SQRT(Vx2+Vy

2) m/s 17.64 17.03 15.44 13.05 10.04 6.80 4.50 5.04 9.65 14.24 17.38 18.47

10 Cy = 2Gm*sin f / V*b 0 0.198 0.378 0.516 0.580 0.495 0.000 -0.667 -0.604 -0.473 -0.335 -0.182

11 a - ao = 10*Cy degree 0.0 2.0 3.8 5.2 5.8 4.9 0.0 -6.7 -6.0 -4.7 -3.4 -1.8

12 R = f(f) 8.5 8.25 7.00 5.25 3.20 1.30 1.00 1.30 3.20 5.25 7.00 8.25

13Ke = 1/ R - 1/ R c -0.01 0.00 0.02 0.07 0.19 0.64 0.88 0.64 0.19 0.07 0.02 0.00

14 ao = f (Ke, a) degree 0.0 0.0 0.0 1.5 4.3 12.3 15.0 12.3 4.3 1.5 0.0 0.0

15 a = (a - ao) + ao degree 0.0 2.0 3.8 6.7 10.1 17.2 15.0 5.6 -1.7 -3.2 -3.4 -1.8

16 g = b + a degree 4.0 17.0 28.9 40.4 49.6 54.7 26.9 -33.5 -50.2 -41.7 -27.3 -10.9


Figure 8. Relative radius of curvature of blade’s trajectories.

Figure 9. Angle a0 in curvilinear flow.


Figure 10. Blade kinematics of vertical axis propellers.

The design blade kinematics is shown in Figure 10. Maximum values of angles g in

the first and second semi-circles are g m1 = 54.7° and g m2 = 50.2°, consequentlyeo = 0.79.The normal blade kinematics according to formula (35) and one of the multi-linkkinematics, are also presented in Figure 10 at the same eo = 0.79.As seen from Figure 10, difference between the design and the used kinematics insome places of the orbit are very great. Consequently, the obtained results can beuseful for designing new regulating mechanisms of vertical axis propellers.

7. Conclusions

A most exact description of a three-dimensional vortex system of vertical axispropeller is presented. A simple method of transformation of free vortices intolongitudinal vortex sheets allows obtaining longitudinal and transverse inductivevelocities at any point of the blade’s orbit. Total velocities induced by vertical andhorizontal (trailing) vortices are uniformly distributed along the blades.Experimental hydrodynamic characteristics of underwater wings in curvilinearmotion are presented, in order to calculate the necessary rotation of propellers bladesabout their attachment axes.The obtained design blade kinematics sufficiently differs from multi-links kinematicsarranged in modern vertical axis propellers. Therefore, the obtained data can beuseful for increasing the efficiency of new vertical axis propellers.


Nomenclature

a relative distance from leading edge of the blade to its attachment axisbp average breadth of the propeller bladeCy dimensionless lift coefficients of the wing profileD diameter of propeller blades orbit (D = 2Rp)k relative wing span (aspect ratio L/b)K relative curvature of the flow (b/R)L length of the blade’s bound vortex (L = 2 Lp)Lp length of the propeller bladeP lift force of the propeller bladeR radius of curvature of blade’s trajectoryR relative radius of the curvature (R/b)R c relative radius of a bent center line of the wing profileu circumferential (rotary) velocity of the blade’s attachment axisV resultant velocity of flow running on the bladeVp velocity of straight line flow running on propellerWx and Wy respectively, longitudinal and transverse inductive velocitiesz number of propeller bladesa angle of attackg angle between blade’s chord and tangent line to the orbitG(f) circulation of the blade’s bound vortexGm maximum circulation of the blade’s bound vortexd relative thickness of the wing profilel advance coefficient of the propellerf angle of the blade’s axis position on the orbits load coefficient of the propellerw angular velocity of the propeller

References

[1] Abbot I.H., Doenhoff A.E., 1959, Theory of wing sections, Dover Publ., NewYork.

[2] Haberman W.L., Caster E.B.,1962,.Performance of vertical axis (cycloidal)propellers according to Isay’s theory, International Shipbuilding Progress, Vol.9, No. 90.

[3] Isay W.H., 1962, Anwendung und Ergebnisse der hydrdynamische TheoriedesVoith-Schneider Propellers, Shiffstechnick, 45 Heft, 9 Band.

[4] Johnson R.W., 1998, The Handbook of Fluid Dynamics, CRC Press, NewYork.


[5] Korn G.A. and Korn T.M., 1968, Mathematical Handbook for Scientists andEngineers, Mc Graw-Hill, New York.

[6] Lavrentyev V.M., 1963, Theory of vertical axis propeller with manyblades,Transactions of Central Marine Investigation Institute, Vol. 49, S.-Petersburg.

[7] Lewis E. V., 1988, Principles of Naval Architecture, vol. 2, SNAME, JerseyCity, N.J.

[8] Loytsyansky L.G., 1978, Mechanics of fluid and gas, Nauka, Moscow.[9] Segal Z.B., 2002, Hydrodynamic Characteristics of Wings in Circular Motion,

Journal of Ship Research, Vol. 46, Number 2.[10] Sparenberg I. A., 1960, On the efficiency of vertical axis propeller, Third

Symposium on Naval Hydrodynamics, Scheveningen.[11] Voytkunsky Y.I., 1985, Handbook of ship theory, Sudostroenie, S.- Petersburg.

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