2 Hydrodynamic characteristics of propeller blades in curvilinear motion. Let, the vertical axis propeller moves ahead with a constant speed V p

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2 Hydrodynamic characteristics of propeller blades in curvilinear motion Practical application of such methods is of some interest for a straight line motion of the wing. But for calculating dynamic characteristics of ship propellers much more complex vortex systems are needed [1,3,6]. Vertical axis propellers have a circular disk which is flush mounted on the horizontal bottom of the ship and rotates about its central vertical axis. On the periphery of the disk five or six vertical blades are arranged with the possibility to rotate about their attachment axes according to definite low which secures the propeller thrust in any necessary direction. Vertical axis propellers proved very effective on tugs and ferries, and are also used at the ship s bow to secure manoeuvering in crowded and restricted waters [7]. The first vertical axis propeller, of Kirsten-Boeing type, was built in 187. Its blades made a half revolution about their axis for each revolution of the whole propeller. A more perfected type of this propulsion, the Voith-Schneider propeller, was introduced in Its blades describe a complete revolution about their attachment axes for each revolution of the propeller disk. Beginning from 196, some serious papers containing hydrodynamic analyses of vertical axis propellers on the base of vortex theory, were published [,3,6,1]. However, most of the authors solved two-dimensional problems; moreover, they did not take into account the great curvature of blades trajectories, or valued the influence of finite aspect ratio of wings and curvature of flow running on the blades not enough exactly. Therefore, often the design results considerably differ from real dynamic characteristics of vertical axis propellers [].. Vortex system and calculation of inductive velocities Let, the vertical axis propeller moves ahead with a constant speed V p in ideal incompressible fluid. The horizontal disk of the propeller rotates about its central axis with a constant angular velocity w. There are z vertical blades arranged on a distance R p from the disk central axis. The root section of the blade is very close to the disk, therefore, according to mirror image [7], the length of a wing s bound vortex L is taken twice of the blade s length L p. As a rule, L p = 1. R p and the average breadth of the blade is b p =.5 L p. Circulation of the bound vortex is G(f) = G m f(f), where G m - 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 = R p. Horizontal section of the propeller, by inverse of fluid motions, is shown in Figure 1, where: u = w R p - circumferential velocity of the blade s attachment axis; W - inductive velocity;

3 Zelik Segal and Alexander Segal 3 V - resultant velocity of fluid running on the propeller blade; P x and P y - respectively, longitudinal and transverse components of the lift force P. Figure 1. Horizontal section of vertical axis propeller. The components P x of all blades form the thrust of the propeller and must be directed ahead in any place on the orbit. Therefore, the blade circulation G(f) in the first (forward) semi-circle ( f p) is directed against the angular velocity w, and in the second semi-circle (p < f < 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 -nd semi-circles, it is necessary that the function of circulation f(f) would be an odd periodical function : f(f) = f( f) (1) Theoretically the most optimal function is f(f) = ± 1, which changes its sign in points f = 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 a jump-in change of the blade s angle of attack in transition points from one semi-circle to another. Therefore it is useful to have a function of type f(f) = sinf, which satisfies the condition (1) and secures enough rapid change of angles of attack in transition points f = and f = p.

4 4 Hydrodynamic characteristics of propeller blades in curvilinear motion By displacement of a blade from location f in location f+ d f circulation, of the bound vortex changes on value dg. Then, according to Kelvin s theorem, a free vertical vortex with circulation - dg is formed in this place. In the first approximation, the propeller is supposed a light loaded, i.e. the inductive velocities are very small, that is why the free vortexes remain immovable, or by inversion of fluid motions they moves with a velocity of running on flow V p. The horizontal (trailing or tip) free vortexes are formed according to Helmholtz s theorem [8, 11]. Consequently, the vortex system of a vertical blade is a set of rectangular closed frames (Figure ). Since G(f) is a continuous function, the frames are continuous vortex sheets moving in the propeller jet. The vortex sheets have a cycloid form according to trajectory of the blade axis (Figure 3): x 1 = V p t + R p sin f = R p (f l + sin f) y 1 = R p (1 cos f), () where t = f / w - time; l = V p /w R p - advance coefficient of the propeller; x 1 and y 1 - respectively, longitudinal and cross axes having a beginning in immovable central axis of the propeller (by inverse motion). Figure. Vortex system of a vertical blade.

5 Zelik Segal and Alexander Segal 5 Figure 3. Cycloidal vortex sheets. Two vertical longitudinal planes, corresponding to f and f + df, are taken in Figure 3. Free vertical vortexes with circulation - dg, disposed between the planes, may be substituted by a continuous vortex layer with circulation: q = - dg(f) / s = -zg m df(f)/p Dl, (3) where s = V p/ z w - distance between free vortexes forming in one semi-circle of the blade s orbit. The sign minus in expression (3) can be dropped by taking real direction of free vortexes (Figure 3). Elements of horizontal vortexes having length dl and circulation G(f), located between above-mentioned two planes, are equivalent to two infinite chains of longitudinal and cross vortex elements with the same circulation G(f) - (Figure 4). The chain of longitudinal vortex elements with length dx 1 = R p (l +cos f)df (Figure 4.b) is approximately substituted by a continuous vortex cord with circulation G( f) zgm f( f)( l + cos f) df Gc = dx1 = s l (4) The chain of transverse vortex elements with length dy 1 = R p sin f df (Figure 4.c) is approximately substituted by a continuous layer with circulation per length q = G ( f )/ s= z G f( f )/ p D l (5) 1 m

6 6 Hydrodynamic characteristics of propeller blades in curvilinear motion Figure 4. Elements of horizontal trailing vortexes. Figure 5. Velocities induced by vertical free vortexes.

7 Zelik Segal and Alexander Segal 7 An arbitrary point A is taken on the blade s attachment axis determined by an angle f. Distance from the point A to upper and lower ends of the bound vortex are, respectively, L 1 and L (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: zgm dgv = qdx = df ( f) dx (6) p D l Velocity induced in point A by the vertical vortex element is determined by known expression based on Biot-Savart formula [8]: dw = 4p dg x v + y (cosq + cos q ) (7) 1 where angles q 1 and q are shown in the Figure 5. Vector of velocity dw is perpendicular to the plane containing the vortex element and point A. Therefore, longitudinal and transverse projections of the velocity dw are equal: dw x = dw y ; dw dw x x + y y = x + y (8) Integrating the expressions (8) on the whole volume of propeller jet and taking in account formula (7) and designations C = z G m /4p Dl; r = x + y + L ; r = x + y + L, the longitudinal and transverse velocities in the point A, induced by free vertical vortexes, will be: W W x y = C =-C y Ú Ú x + y b x Ú Ú x + y b L1 L ( + ) df ( f) dx (9) r r 1 L1 L ( + ) df ( f) dx (1) r r 1 where b = R p (sin f sin f); y = R p (cos f cos f ) (11) An area on the upper surface of propeller jet with sides dx and dy, according to Figure 4, has a longitudinal vortex element with circulation G c and cross vortex element with length dy = R p sin f df and circulation G 1 = q 1 dx (Figure 6). 1 1

8 8 Hydrodynamic characteristics of propeller blades in curvilinear motion Figure 6. Velocities induced by horizontal free vortexes. The longitudinal velocity induced by cross vortex element in the point A, according to Biot Savart formula [1, 8], is: dw 1x G1dy sin = u cos j 4pr 1 (1) where u - angle between the vortex element and radius-vector r 1 connecting the vortex element with the point A (Figure 6); sinu = n 1 ; cosj = L 1 ; sinucosj = L 1. r1 n1 r1 Similar formulas with values r and L correspond to lower surface of the propeller jet. The longitudinal element of upper vortex surface with length dx and circulation G c induces a transverse velocity in the point A (Figure 6). Projection on axis y of this velocity is equal: dw 1y Gcdx sin =- b cos V 4pr 1 (13)

9 Zelik Segal and Alexander Segal 9 where sin b = l 1 /r 1 ; cos x = L 1 /l 1 ; sin b cos x = L 1 /r 1. Integrating the expressions of type (1) and (13) about the upper and lower surfaces of the propeller jet gives the longitudinal and transverse velocities in the point A, induced by free horizontal vortexes: W W 1x 1y L1 L = CRp Ú Ú ( + ) f( f)sinfdf dx (14) r r b L1 L =- CRp Ú Ú ( + ) f( f)( l + cos f) df dx (15) r r b Taking the integrals (9), (1), (14) and (15) on x (see [5] - table integrals No. 146, 6 and 67) gives: L1 L bl1 bl Wx = C Ú (arctan + arctan -arctan -arctan ) df ( f ) (16) y y yr yr r L r L W y =- 1 C Ú ln ( )( ) ( r -L )( r -L ) df ( f ) (17) W W = CR Ú 1x p =-CR 1y p L1 b L b [ ( 1- ) + f d y L r ( 1- )] ( f)sinf f (18) + y + L r Ú 1 1 L1 b L b [ ( 1- ) + f d y L r ( 1- )] ( f)( l + cos f) f (19) + y + L r 1 1 where r 1 and r contain, instead of x, the lower limit of integrals b. Numerical integration of expressions (16) (19) doesn t present any difficulties and allows to determine longitudinal and transverse velocities induced by free vortexes in any 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 will be considered a two-dimensional problem: L 1 = L =, consequently, W 1x = W 1y = and r 1 L 1 ª.5 (b +y )/L 1. Taking in account (1) and Ú df ( f) =, equations (16) and (17) are reduced to:

10 1 Hydrodynamic characteristics of propeller blades in curvilinear motion b Wx = C Ú ( p signy - arctan y ) df ( f) () Wy = C Ú ln( b + y ) df ( f ) (1) When the point A(f ) is in the first semi-circle ( f p) the function sign y = (+) by integration anti-clockwise from -f up to f, and equal to (-) by integration from f to p - f Figure ( 7.a). When the point A(f ) is in the second semi-circle (Figure 7.b), the function sign y = (+) by integration anti-clockwise from f up to - f, and (-) from - f up to f. Figure 7. Design point A in 1-st or in -nd semi-circle. Taking in account the condition (1), the first part of integral () is reduced to: -f 1. f p: pc Ú signy * df ( f) = pc Ú df ( f) - pc Ú df ( f) = 4pCf ( f) f -f f () -f. p < f < p: pc Ú signy * df ( f) = pc Ú df ( f) - pc Ú df ( f) = -4pCf ( f) f f -f (3)

11 Zelik Segal and Alexander Segal 11 For the second part of integral () it is used follow transformations: b sinf - sinf p f + f p f + f arctan = arctan = -arc cot(cot ) = - y cosf - cosf f + f For main value of arccot it is: < < p, that gives: - f < f < p - f, consequently: -f - - C Ú arctan b df = - Ú - = - - Ú + y ( ) C ( p f f f df C df C ) ( f ) ( p f ) ( f ) Ú fdf ( f ) -f -f -f -f The first integral in the right part of the last expression is equal to zero since f(-f ) = = f( - f ); 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: -f arctan b y df ( f ) C f f ( f ) C f ( f ) d f p Cf ( f ) (4) -f - C Ú = [ ] - Ú = - -f -f Adding the expressions () and (3) with (4) and confining it by f(f ) = sin f gives: 1. f p: W x = C sin f (5) -f. p < f < p: W x = 6p C sin f (6) In the second semi-circle sinf <, therefore in all points of the orbit the longitudinal inductive velocities W x are directed as the flow velocity V p. And in the points symmetrical about cross diameter, longitudinal velocities in the second semi-circle are three times more than in the first semi-circle. For calculation of the transverse inductive velocities, the values b = R p (sin f sin f) and y = R p (cos f cos f ) are substituted in (1). Then, after simple transformations and taking in account the condition (1) the expression (1) is reduced to: f - f Wy = C Ú lnsin df( f) (7) Taking f(f) = sin f and integrating (7) by parts gives:

12 1 Hydrodynamic characteristics of propeller blades in curvilinear motion Wy =- - C Ú sin f f f cot d f (8) f - f Addition of integral Ú -sinf cot d f = to (8) and taking in account that f - f cosf + cosf cot =, allows to obtain a final formula for transverse velocity at sinf - sinf L = :Wy =- pccos f (9) 4. Numerical analysis of inductive velocities Numerical integration of expressions (16) (19) was made for G( f) = G m sinf, l =.6 and L p /D =.6 at various positions of propeller blades on the orbit and some points along the blade axes. Calculation results have shown that inductive velocities W x, W 1x, W y and W 1y each separately are distributed along the blade not uniformly; but sums W x + W 1x and W y + W 1y are constant along the blades with high exactness, except a very small part (-3%) of the blade length, near the end of the blade. zgm Table 1 presents calculation results of inductive velocities, divided by C = 4p Dl, for finite blade length and also for -dimensional problem according to formulas (5), (6) and (7). As seen from Table 1, difference between velocities of finite and infinite blades is only 4 1 % for longitudinal velocities, and about 5 9 % for transverse 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, in particular, to compare them with the velocity of running on flow V p. Table 1. Dimensionless inductive velocities at G(f) = G m sin f. f, deg W x +W 1x W x (-dim.) W y +W 1y W y (-dim.) The lift force of a bound vortex is determined according to formula of Joukowsky [8]:

13 P Zelik Segal and Alexander Segal 13 = rvl p G ( f) (3) where r = 1 kg/m 3 r is the water density. Since the force P is directed perpendicular to resultant velocity V, the longitudinal components of it: V p, W x and W 1x do not create the propeller thrust, and the transverse inductive velocities are nearly symmetric about longitudinal diameter of the propeller. Velocities induced by bound vortexes are relatively small. Consequently, for calculation of the propeller thrust T it can be used only the cross component of rotary velocity of the blade usin f : zrgmulp T = Ú sin df = 5. GmrzuLp (31) For example, it is taken a vertical axis propeller with follow parameters: D =. m, L p = 1. m, b p =.3 m, a =.3, z = 6, n = 15 r.p.m, u =.5 D w = 11. m/s, V p = 6.6 m/s, l = V p /u =.6, T = 4 N, s = T/r D L p V p =.765. Then, according to (31), the value of maximum circulation will be G m = 1. 1 m /s. zgm After determination of C = 4 =.18 m/s, the dimensional values of all p Dl inductive velocities can be calculated according to data of Table 1. For example, the maximum longitudinal velocities in 1st and nd semi-circles are, respectively, W m1 = =.7 m/s, and W m = =.51 m/s. These values are enough great relatively to flow velocity V p = 6.6 m/s. In conclusion of this part, it must be noted, that in formulas (4) - (6) the distance V between free vortexes s = p z v is taken on assumption that by motion of the propeller with velocity V p the free vortexes remain immovable (or by inversion of motion they move with the flow velocity V p ). But in reality they move owing to presence of inductive velocities. It can be approximately taken a conditional increase of the velocity V p by a some value W c. If this value will be equal to a half sum of average longitudinal inductive velocities on the first and second semi-circles of the orbit, then at s =.5.8 the design inductive velocities will be decreased on 1-15 %. 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 the vertical free vortexes lift forces directed against the velocity V p. All of these factors allow to consider the decrease of obtained inductive velocities at s =.5.8 not more than 7 1 %.

14 14 Hydrodynamic characteristics of propeller blades in curvilinear motion 5. Experimental hydrodynamic characteristics of blades If vectors of flow velocities at leading and trailing edges of a wing do not directed on one line, then the flow about the wing is curvilinear. Such a flow along the profile occurs, in particular, when a longitudinal axis of a wing moves along a curvilinear trajectory. According to equations (), relative radius of the trajectory s curvature of a propeller blade R = R/ b on dependence of angle f and advance coefficient l is presented in Figure 8 at b =.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 real fluid, even by small angles of attack, essentially differ from analogous characteristics in straight line flow. Therefore, experimental characteristics of wings in curvilinear motion are necessary for design of vertical axis propeller. They are also a methodical interest in general theory of wings. The experimental determining of wing hydrodynamic characteristics was carried out in a circular testing basin of S.-Petersburg Water Transport Institute. Four models of wings with a chord b = 3 cm and length L = 4 cm were tested. Three of the models had a symmetrical profile NACA with different relative thickness d:.1,.15 and.. The profile of the fourth model was derived from NACA-15 section shape by bending its axis on radius R c = 6. 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 was approximately 3 mm, and the upper part of the model pierced the free surface of water. 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 =.45 m/s. The corresponding Froude number V/ gb =.6 secured a practically waveless mode of the wing motion. Consequently, the basin bottom and the free surface of water can be considered 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 of attack of propeller blades. An important parameter for hydrodynamic characteristics of wings in curvilinear motion is a distance from attaching axis O to the leading edge A. In the experimental data and calculations it is used a relative value a = AO/b. The experiment was carried out at a =.45, but all the results can be simple converted to any value of a [9]. C y It is known that derivative a and angle of zero-lift direction a practically do not depend on the Reynolds number [8, 11]. Therefore, values of a and C y a obtained in experiment at Re = are available too for the full scale conditions. Moreover, the experimental data were compared with known results of wind tunnel tests performed in straight line flow [1, 11]

15 Zelik Segal and Alexander Segal 15 For converting this results to infinite wing span, the known following formulas have been used [8, 11]: Cy Bk C y ( ) k = ;tana a + i =, (3) k pk where C y ( ) k - derivative corresponding to tunnel tests of wings with a finite span k; a B - derivative corresponding to infinite span; a i - down wash angle between lift force and perpendicular to wing chord. According to theoretical investigations, B =, but in real fluid for d =.1.15 according to tunnel tests and experiments in curvilinear wing motion, it is approximately A = Therefore, when the effective angle of attack is in degree, then C y =.1 (a a o ) (33) where C y - lift coefficient for infinite wing span; a - angle of attack between profile chord and resultant velocity of water V. The maximum lift coefficient C ymax increases, as a rule, about 3 % by increase of the Reynolds number from 1 5 to 1 7 [6, 7]. In any case, by design of vertical axis propellers the effective angles of attack, as a rule, does not exceed 8 1 o, which are displaced in the straight line part of the curves C y = f (a a o ). Angles of zero-lift direction a in dependence from relative curvature K = 1/R and relative distance a are presented in Figure 9. Analysis of the hydrodynamic characteristics of the fourth model has shown that the bend of the profile center line is equivalent to a corresponding curvature of the wing s motion. Therefore, to use experimental results, an equivalent curvature of flow must be calculated by following formula K e = 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 g between profile chord and tangent line to the blade s orbit. Dependence g = f(f) is named as blade kinematics.

16 16 Hydrodynamic characteristics of propeller blades in curvilinear motion In the first Foith-Schneider propellers a normal kinematics was used, when all perpendiculars to the blade s chords have crossed in one point regulation pole. Ratio of the pole eccentricity to orbit radius R p is named as relative eccentricity e o. Normal kinematics is expressed by dependence sinf g = arctan 1/ e o + cosf (35) According to formula (35) sinus of maximum angle g is equal to e o. Therefore, for any dependence g = f( f) relative eccentricity is determined conditionally as m + m e o = sin ( g 1 g ), where g m1 and g m 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-link mechanism, where each blade is connected by several links and hinges with one regulating pole. The value e o in such kinematics achieves In test models of vertical axis propellers, as a rule, a peripheral cam imparts any desired motion of the blades. And there are projects to use cam mechanisms also in regular 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]: r y p p P = C V Lb (37) Equating the right parts of (3) and (37) gives: C y = G ( f)/ Vb p (38) Calculation of blade s kinematics for above-mentioned example of vertical axis propeller is presented in Table. In first rows of the Table there are longitudinal and transverse inductive velocities decreased (in comparison with data of Table 1) by 1%, 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 b between the velocity V and tangent line to blade s orbit according to 1 values of the angles f. Lift coefficient of the blade C y is calculated in row 1 according to formula (38), taking in account G( f) = G m sinf and G m = 1.1 m /s. The effective angle of

17 Zelik Segal and Alexander Segal 17 attack is calculated by formula (33) as for infinite wing span, since the inductive velocities were defined by means of a three-dimensional vortex system. Relative radiuses of flow curvature are obtained from Figure 8, and only in the point f = 18 it is taken R = 1.. Equivalent curvature of the flow K e for blades with a bent profile center line is determined on expression (34). The angle of zero-lift direction a is taken from Figure 9 by a =.3. The angles g are negative when the leading edge of the blade is inside of the orbit. Table. Calculation of blade kinematics at G(f) = G m sin f. 1 f degree W x +W 1x m/s W y +W 1y m/s V x = V p +Ucosf+W x +W 1x m/s V y = Usinf+W y +W 1y m/s tan q =V y / V x q = Atan q * 57.3 degree b = f - q (-18 or -36) degree V = SQRT(V x +V y ) m/s C y = G m *sin f / V*b a - a o = 1*C y degree R = f(f) K e = 1/ R - 1/ R c a o = f (K e, a) degree a = (a - a o ) + a o degree g = b + a degree

18 18 Hydrodynamic characteristics of propeller blades in curvilinear motion Figure 8. Relative radius of curvature of blade s trajectories. Figure 9. Angle a in curvilinear flow.

19 Zelik Segal and Alexander Segal 19 Figure 1. Blade kinematics of vertical axis propellers. The design blade kinematics is shown in Figure 1. Maximum values of angles g in the first and second semi-circles are g m1 = 54.7 and g m = 5., consequently e o =.79. The normal blade kinematics according to formula (35) and one of the multi-link kinematics, are also presented in Figure 1 at the same e o =.79. As seen from Figure 1, difference between the design and the used kinematics in some places of the orbit are very great. Consequently, the obtained results can be useful for designing new regulating mechanisms of vertical axis propellers. 7. Conclusions A most exact description of a three-dimensional vortex system of vertical axis propeller is presented. A simple method of transformation of free vortices into longitudinal vortex sheets allows obtaining longitudinal and transverse inductive velocities at any point of the blade s orbit. Total velocities induced by vertical and horizontal (trailing) vortices are uniformly distributed along the blades. Experimental hydrodynamic characteristics of underwater wings in curvilinear motion are presented, in order to calculate the necessary rotation of propellers blades about their attachment axes. The obtained design blade kinematics sufficiently differs from multi-links kinematics arranged in modern vertical axis propellers. Therefore, the obtained data can be useful for increasing the efficiency of new vertical axis propellers.

20 Hydrodynamic characteristics of propeller blades in curvilinear motion Nomenclature a relative distance from leading edge of the blade to its attachment axis b p average breadth of the propeller blade C y dimensionless lift coefficients of the wing profile D diameter of propeller blades orbit (D = R p ) 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 = L p ) L p length of the propeller blade P lift force of the propeller blade R radius of curvature of blade s trajectory R relative radius of the curvature (R/b) R c relative radius of a bent center line of the wing profile u circumferential (rotary) velocity of the blade s attachment axis V resultant velocity of flow running on the blade V p velocity of straight line flow running on propeller W x and W y respectively, longitudinal and transverse inductive velocities z number of propeller blades a angle of attack g angle between blade s chord and tangent line to the orbit G(f) circulation of the blade s bound vortex G m maximum circulation of the blade s bound vortex d relative thickness of the wing profile l advance coefficient of the propeller f angle of the blade s axis position on the orbit s load coefficient of the propeller w angular velocity of the propeller References [1] Abbot I.H., Doenhoff A.E., 1959, Theory of wing sections, Dover Publ., New York. [] Haberman W.L., Caster E.B.,196,.Performance of vertical axis (cycloidal) propellers according to Isay s theory, International Shipbuilding Progress, Vol. 9, No. 9. [3] Isay W.H., 196, Anwendung und Ergebnisse der hydrdynamische Theorie desvoith-schneider Propellers, Shiffstechnick, 45 Heft, 9 Band. [4] Johnson R.W., 1998, The Handbook of Fluid Dynamics, CRC Press, New York.

21 Zelik Segal and Alexander Segal 1 [5] Korn G.A. and Korn T.M., 1968, Mathematical Handbook for Scientists and Engineers, Mc Graw-Hill, New York. [6] Lavrentyev V.M., 1963, Theory of vertical axis propeller with many blades,transactions of Central Marine Investigation Institute, Vol. 49, S.- Petersburg. [7] Lewis E. V., 1988, Principles of Naval Architecture, vol., SNAME, Jersey City, N.J. [8] Loytsyansky L.G., 1978, Mechanics of fluid and gas, Nauka, Moscow. [9] Segal Z.B.,, Hydrodynamic Characteristics of Wings in Circular Motion, Journal of Ship Research, Vol. 46, Number. [1] Sparenberg I. A., 196, 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.