On the Vortex Theory of Screw Propellers

On the Vortex Theory of Screw PropellersAuthor(s): Sydney GoldsteinSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 123, No. 792 (Apr. 6, 1929), pp. 440-465Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/95206 .

Accessed: 08/05/2014 10:12

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Society of London. Series A, Containing Papers of a Mathematical and Physical Character.

http://www.jstor.org

This content downloaded from 169.229.32.137 on Thu, 8 May 2014 10:12:45 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/95206?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


4440

On the Vortex Theory of Screw Propellers. By SYDNEY GOLDSTEIN, M.A., Ph.D., St. John's College, Cambridge, and the

Kaiser Wilhelm, Institut fflr Strdmungsforschung, Gdttingen.

(Communicated by L. Prandtl, For.Mem.R.S.-Received January 21, 1929.)

1. Introduction.

The vortex-theory of screw propellers develops along similar lines to aerofoil theory. There is circulation of flow round each blade; this circulation vanishes at the tip and the root. The blade may be replaced by a bound vortex system, which, for the sake of simplicity, may be taken, as a first approximation, to be a bound vortex line. The strength of the vortex at any point is equal to F, the circulation round the corresponding blade section. From every point of this bound vortex spring free, trailing vortices, whose strength per unit length is -alP/ar, where r is distance from the axis of the screw. When the interference flow of this vortex system is small compared with the velocity of the blades, the trailing vortices are approximately helices, and together build a helical or screw surface.* Part of the work supplied by the motor is lost in producing the trailing vortex system. When the distribution of P along the blade is such that, for a given thrust, the energy so lost per unit time is a minimum, then the flow far behind the screw is the same as if the screw surface formed by the trailing vortices was rigid, and moved backwards in the direction of its axis with a constant velocity, the flow being that of classical hydro- dynaniics in an inviscid fluid, continuous, irrotational, and without circulation.t The circulation round any blade section is then equal to the discontinuity in

* This system is unstable, and at a sufficient distance behind the propeller the vorticitv is mainly concentrated into as many helical vortices as there are blades, with radii somewhat less than that of the original syrstem, and into a straight line vortex along the axis. The strength of each helical vortex is nearly equLal to the maximum value of the circulation round a blade section, and the strength of the straight line vortex is equal and opposite to the total strength of all the helical vortices.

t Betz, 'G5ttinger Nachr.,' pp. 193-213 (1919); reprinted in 'Vier Abhandlungen zur Hydrodynamik und Aerodynamik,' L. Prandtl und A. Betz, Gottingen, 1927, where a selected bibliography is given. The theory takes no account of the finite size of the boss, or of the influence of compressibility of the fluid at large tip speeds. The approximation that the trailing vortices form regular helices is equivalent to neglecting the contraction of the slip stream, and is valid only for small values of the thrust coefficient. Also, in finding the most favourable distribution, the energy losses arising from the profile drag of the blade sections are not taken into account.



Vortex Theory of Screw Propellers. 441

the velocity potential at the corresponding point of the screw surface. Further, for symmetrical screws, the interference flow at the blade is half that at the corresponding point of the screw surface far behind the propeller.*

An approximate solution for the irrotational motion of a screw surface in an inviscid fluid was given by Prandtl.t The accuracy of the approximation increases with the number of blades and with the ratio of the tip speed to the velocity of advance, but for given values of these numbers we have no means of estimating the error, since the exact solution of the problem has not yet -been found. It is theJmain object of this work to find the exact solution.

We consider first the important case of the two-bladed propeller; formulhe for any number of blades are given later. Finally, the application of the results is briefly considered.

2. Formulatioon of the Potential Problem.

Let R be the radius, o the angular velocity, and v the velocity of advance of a two.bladed propeller. Let e be the angle between 0 and -17t whose tangent is v/ro, where r is the distance from the axis of rotation. The equation to the screw surface to be considered is

0-cz/v _ O or , 0O r?IR, (1)

where r, 0 anid z are cylindrical polar co-ordinates. The axis of z is along the axis of the screw surface, with its positive direction away from the propeller.t If w is the velocity of the screw surface in the direction of its axis, and k the potential, such that grad 4 is the fluid velocity, the boundary conditions are that

w cos C -cos s -aa sin , (2)

or

cortv = . (3)

for 0 - coz/v 0 or -r and 0 < r KR, and that grad b should vanish for r = oo.

The geometrical conditions are such that the fluid velocity is a fuiction of r and 0 -- oz!v only. This does not by itself restrict f, to be a function of r and 0 - wz/v only, since constant multiples of 0 only and of z only may

* Betz, loc. cit. t ' Gttinger Nadhr.,' pp. 213-217 (1919); reprinted in the 'Vier Abhandlungen.' $ Since we are considering the motion far behind the propeller, the screw surface may

be taken as extending to infinity in both directions along the axis of z.



442 S. Goldstein.

occur. But no term in 0 or z may occur in the expression for 'f for r > R, since the second would gi-ve finite velocity for infinite r, and the first would

give circulation round the axis. Conditions of continuity now suffice to show that no term in 0 or z may occur in the expression for f for r < R. Hence j is a function of r and C, where

0 - (OZ/V. (4)

The boundary condition (3) then becomes

ar = - WS + 'CI)22XS X (5) v2 + C02r2 co

for =0or x and 0 < -r R.

It is convenient to take wvl/ temporarily equal to 1. Also, let

e wr/v. (6) Then (5) becomes

_2

aX - - 2 (7}

for 0 0 or i, and 0 6 r < R. The differential equation satisfied by i namely V2= 0, can then be -written

/3)2 32 (8)

In addition to satisfying (7), must be a sinigle-valued function of position, aiid its derivatives must vanish when r is infinite. It must also be continuous everywhere except at the screw surface.

3. Solution of the Potential Problem.

3.1. The Formt of thte Sotution.*--It is not difficult to see that the conditions of the problem-are such that q is an odd function of 4 and of {Wr - h, and since it is a single-valued function of position, conitinuous for r> RI it can be expanded, for r > R, in a series of sines of even multiples of C. Assumng siich an expansion, diferentiating term by term, and substituting in 2 (8), we find that the coefficient of sin 2nl must be a linear funetion of '2 (2n[t) Pand K2, (2n ), where 1,, and K,, are the modified Bessel functions. But

* The reasons for adopting the form and method of solution to be described will be elear to anyone who reads Appendices I and 2.



Vortex Theory of Screw Propellers. 4t43

12' (2n ) cannot occur, since grad b must vanish when r, or t, is infinite. Hence we may assume

+XK2"(2n4 sin2n , for r>P (1) n 1 K2. (2npo)

where, the c are conistants to be determined,

G= oR/v, (2) . .~~~~~~l

and K2n, (2n V) has been inserted in the denominator for future convemence. We shall assume that the expansion holds also for r R. For 0 ? r < R, we put

+ - rp2 + (3) so that

+ (1 + 02)V1 = (.Ad)2( 82)4 (4)

and a= O ~~~~~~~~~~(5)

at 4 0 and it. Consider first the region for which 0? < <ri. For this region

it 4 oo cos (2m +1) ( 2=2 - =O (2m+1)2

If we assume 40

differentiate term by term, and substitute 'in (4), we find that

fof) 1- (d ) ( j)' (8) and

(V) (m 1)2 (1 ?.0 ~2fm) (92+)2))It4 (d-p)~ ~ ~ ~ ~~~~~~7 .(2m(2fi +)( U2g ")=_ 12 4t \1 +2 + must be finite at r = 0, and so contains no term in log .. Hence

]fo (W.L-)12 [,O2/(1 + ",2). (10) If we put

fm ( UL k 2 j) {g (L))(1 te (2m n1) 1 + t (

then

(" 1 qwl(j) -(2m + 31)2 (I + to) g,, t~) = (2 m +1l)2 to* (12)



444 S. Goldstein.

a particular solution of which is S1, 2m+ 1 (2m + 1 i ), where S,,, (z) is a Lomme function as defined in Watson's ' Bessel Functions,' ? 10.71, p. 347. The

general solution contains, in addition, terms in 12m+-1 (2m + 1 St) and K2m+1

(2m + I 4). Corresponding to equation 10.71 (3) of Watson's 'Bessel Functions' we

have

8SI (iz) 2 sin 1 1, (z) + v e K, (z)-t1,, (z) (13)

where

z2 z4 6 t,(z) + + 2-v2 (22-v2) (42-V2) (22 'v2) (42 V2) (62 -V2)?.*

(22 -V2) (42-v2) .. (4M2 - v2)?. (14)

In order to have a function which is real when z is real, and which (for positive sv) has no singularity at z = 0, we define the function T1,, (z) as being equal to the expression on the right of (13) without the term in K,, (z). Thus

Tj,V (z) = 2 W r Iv (z) -tj., I (z) (5

2sin -lv7r

where ti, (z) is given by (14).* Then T1, 2m-+ (2rn- + 1 ) is also a particular solution of (12). Its value is found by putting (2mn + 1) for v and (2m + 1) &- for z in (14) and (15). As the general solution of (12) which remains finite on the axis, when t is zero, we find then

g.(p) T1= ,2a+1 (2m + 1V,) + b11, '2,m+1 (2m + 1I), (16)

where the bm are constants to be determined. If now in (3) we substitute for 4 from (6), for 0 from (7), forfo from (10)

and for f r fom (11), we find

7c m-o (2E r + 1)2

4 [4 Tl. 2,+, (2m + I [) + I2in+ (2m + Il) cos (2m + i1) (17) m=O - (2n + 1)2 2m+1 (2m? + 1 ti)

where the am are new constants to be determined, and V.( is coR/v, as before. This expression holds for 0 < r < R, and for 0 < 7. Also 0 is an odd

* The expression given in (15) for T1, , (z) is indeterminate when v is an even integer. This happens in the case of the four-bladed propeller, anid will be discussed later.




function of C, and for -- < < 0, we have the same expression with the opposite sign. The discontinuity in I at 4 = 0 or nr is thus

b] 8 S Ti.2m+ (2m + 1) 7rmO (2m+ 1)2

+ c amI2m+1 (2m +1) (18) m=O I2 +1 (2m + I Vo)

It remains to determine the constants am in (17) and cm in (1) from the conditions of continuity at r R. But to do this we shall require to have the numerical values of the T functions and their derivatives. These are not tabulated, and so we must find formulae for computing them. It will be convenient to discuss at the same time the numerical evaluation of the first term of (18), which is independent of pt. This term gives the result for infinite p.0, that is, infinite R; the second term gives the influence of the edge.

3.2. Discussion of the T Functions.-The formule defining T, given in 3.1 (14) and (15) (p. 444), are suitable for computation for small ,t. Again, Ti, . (z) and SI, (iz) differ only by a multiple of K, (z), which is exponentially small for large values of the real part of z. Hence for I arg z 12X, Tj, (z) has the same asymptotic expansion as Sl.> (iz), namely

Ti, . (z) 1

-, V2/Z2 + M2(v2 22)/Z4 - V2 (2 22) (v2 -42)/Z6 - (1)

(Watson's ' Bessel Functions, ?10.75, p. 351). Another formula for T, 2.m+1 (2m + 1 ) can be found by substituting

T,.2m+ (2m + l) = -o ( p) + r2 ([,)1(2m + 1)2 + T4 ( )/(2m + 1)4 +- ... (2)

in the equation 3.1 (12) and equating to zero the coefficients of powers of (2m + 1). This gives

170 =2/(j + U2), (3) and

tr+2 2(Y) () so that

2 4 l2 (1 2)/(1 + (,2)4)

=16{1- 142 + 219 -

4 V6}1(1 + p2)7, (6) and

= 64 2 {I 75 [t2 + 603 4 -'1065 (l6 + 460 0 - 36 ,u 10 }1(1 + 42)10. (7)



_446 S. Goldstein.

(The expansion is probably convergent for some values of ,uo and asymptotli for others.) It is not immediately obvious that the formal solution of 3.1 (12) so obtained represents Tl,2m+1 (2m + I t); that this is so becomes clear on comparing numerical values calculated from 3.1 (14) and (15), or from (1) above, with those calculated from (2).

By using the formuhle 3.1 (14), (15), 3.2 (1), (2) to (7) we can now tabulate T2 2,,+, (2m + 14 We find that it is always very near to 02/(1 + t2),

and for any given value of t approaches nearer and nearer this value as m inc:eases. If we put

1 ? 2 Ti-,2m+i (2n + I ,) = , ++tL(p}) (8)

the values of F2m+ ( G) for various values of m and ,u are given in the table below.

Table T.

| F, ( S4). F3 ( Si F ) G( )

0-2 0-0*1060 -0*156 - 0055 01260 0-4 I -0-1287 -0-0271 -0*0110 0-2450 0-6 -0-1052 -0-0231 -0 0096 0-3524 0-8 -0-0656 -0-0125 -0 0049 0-4447 1-0 -0-0316 -0-0026 -0-0005 0-5259 1-2 -0-0039 +0-0040 ? +0-0021 0-5930 1-4 +0-0140 +0-0075 0-0034 0-6501 1-6 +0-0251 0-0087 0-0036 0-6979 1-8 0-0311 0-0085 0-0034 0-7381 2-0 0-0336 0-0080 0-0030 0-7720 2-5 0-0316 0-0048 0-0020 0-8360 3-0 0-0254 0-0035 0-0011 0-8791 3-5 0-0192 0-0021 0-0007 0-9087 4-0 0-0141 0-0014 0-0005 0-9296 4-5 0-0102 0-0009 0-0003 0-9446 5-0 0-0073 0-0006 0-0002 0-9555 6-0 0-0039 0-0003 0-0001 0-9698 7-0 0-0021 0-0002 0!0001 0-9783 8-0 0-0012 0-0001 - 0-9836 9-0 0-0007 0-0001 - 0-9872

The last figure in the entries in this table mnay be incorrect.

Then

8 ~ T1 2m?i(2m + ItL) 8 $ GO 1 8 G F2m+l) 7 O (2m=o+ 1) 7 1+ - mo(2m+1)2 mC _o=(2m +1)2

1 + 2 72 0 (2m + )2 (9)




The expression in curly brackets is also tabulated in Table I, where it is denoted by G (,u). Since the F are small, and decrease with m, the tabulation is rapid.

The values of G(V.) for V = 2-8, 3 8, 4 8, 55, 5, 8, 6 5, 6 8, 7 5, 7 8, 8 5, 8 8, 9-5 and 9-8 will be required later. They were foLnd by interpolating in tables of [2/(1 + 02) - G (4), etc.

This completes the discussion of the first term of 3.1 (18) (p. 445). To evaluate the second termn we shall require the values of the derivatives of the T functions, formulhe for which are obtained by differentiating the formula@ for the T functions themselves. The approach of (2m + 1) T'1i 2rn? (2m + 1 t) is to 2t/(1 + t2)2

A few values are given below for exhibition purposes. (The dash denotes differentiation, so that T'1,,, (z) is dT1, (z)/dz).

T', 1 (5) - -0O185, 3T'1.3 (15) =0 0152, 2 . 5/(1 + 25)2 - 00148.

T', 1 (3) = 0 0723, 3T'1 3 (9) -=0 0630, 2 . 3/(1 + 9)2 = -0600.

T'1,l (2) = 3 61543, 3T'13 (6) O*1649, 2 . 2/(1 + 4)2 01600.

3.3. The Determination of the Constants am.-We must now turn to the determination of the constants a. in 3.1 (17) and (18) (pp. 444-5) from the conditions of continuity at r =R. The conditions are the continuity of b and of ao/ar. At r R, and 4 = 0 or n, singularities are to be expected, and so we cannot obtain analytical conditions for continuity there. Also + is an odd function of C. Hence it is sufficient to ensure continuity for 0 <1 < n, and continuity for -7 <E <0 will automatically follow. But in the range 'o < 4 < 7,

8 co n cos (2rnm+ 1) = n = 14n2 _ (2m +1)2 (1)

We substitute this into 3.1 (17), re-arrange the resuLlting double series, and equate the coefficient of sin 2nG so obtained to the coefficient of sin 2nG in 3.1 (1) (p. 443), for V = ,>. We then differentiate both 3.1 (1) and 3.1 (17) term by term with respect to ,t and repeat the process. This gives us the equations

_ _L1 2 + +ai - r_ I(2)

m07z (2m +1)2 ; 4n2 _(2m +l)2 and

4 T'1 2m+1 I'2m?l 4 (2m + 1) K'2n 3 C l~~9+ amT i~-7C 3

m -7 (2m +1 am12m+1} 4n2 (2m+1)2 K2

The argument of the. functions Ti.2m+i, T'"2m+l, '2m+1' and 1'2m+1 is (2m + 1) 0o; that of K2, and K'2n is 2n,u0. Dashes denote differentiations,



448 S. Goldstein.

as before. Both equations hold for all integral vdAlues of n. Eliminating c gives us the equation

'2m+IK -

am {(2m8 + 1) 1+1 K2n 7Z 1~~2m+1 I2

4m o 4n2- (2m + 1)2 o

TIT 2m+1 (2m 2+1)TI.2m?1 ( wtK2,az -o (2mt +1)2 [E4n2 (2m + 1)2] o(2m+1)2[4n2- (2m+1)2]

We have then an infinite number of linear equations with an infinite number of unknowns. The standard method of solving is to find successive approximations-to solve the first equation with a1 = a2 a3 = .. 0, then the first two equations with a2 = a3 ... 0, and so on. But we may anticipate from the singularity in + at the edges (r = R, 4 = 0 or -t) that the convergence will be very slow. 'We have to solve for a whole set of values of [,t and the work would be prohibitive and the results inaccurate. Fortunately we can escape this difficulty. - __

We have seen that, so long as 0t is not too small, Ti,2m?l (2mn + 1l,u) is nearly equal to uo21 + so2), and that (2mn + I)T'1 2m+1 iS small compared with T1.2.?+l; also (2m + 1)1"2wI-1 /12.?1 - 2-K'2n/K2n is nearly equal to (I + ,tP (2m + I + 2n), and K'2E/K2,, is nearly equal to -(1 + [o2)l.*

Thus approximately our equationis may be written

am =- 2n o- io 2 _

j_2_ 4 _ _2

_ _ _ 1_2

4 m=O 2n - 2m -I f + o (2m + 1)2 [4n2 - (2 4 +m)2 ] - A2 2

7-2 (5)

I + 2L02 16n or

v aX7T, ,,, = _A0Si:,, a; ~~~~(6) 2i 2 - 2-n - I I O2 4n(

These equations may be solved exactly (Appendix 3), the solution being

am ' _ 2 A),,, where i + 0

A =1, 3A1 =-, 5 I2 3 2 I .3 - - (7)

We then putt 2

aM - oL 2Am + n,) (8)

* See the asymptotic approximations to IX (nx) and K,, (nx) given by Nicholson, 'Phil. Mag.,I vol. 20, p. 938 (1910). But the approximation r much better than might be expected, especially for (2m + l) 1'"2r1/'2m+1- 2nK'2n/K2n.

t The symbol F is diff orently used in section 2.



Vortex Tlheory of Screw Propellers. 449

substitute in (4), and solve numerically for e 0, e, ... in the manner described above. The x are small; even for the lowest value of VO to be considered, namely 2, the degree of accuracy we shall finally aim at is attained if zo is found correct to two significant figures, with a possible error of 1 or 2 in the second figure, if el is found to one significant figure, and if the other s are ignored. The convergence is slow, but not so slow that we cannot manage this with a reasonable amount of labour. The equations -were solved numerically for to = 2, 3 and 5, with the following results

o-2; 02 -0-061, rj0013. (9)

- = 3, zo = 0 -047, r1 = 0(07. (10)

lo -5; so - 0 033, s1=0004. (11)

It will be seen that, to one significant figure, -e is 0 06 when ,uo is 2; it is ( -05 when ,uo is 3, and 0 03 when V, is 5. It was therefore taken, without any further ado, to be 0 04 when pt is 4; to be 0 02 when p0 is 6, and 0 01 when ,uo is 7; and the other s were neglected for these values of pt.* For higher values of suo all the ? may be neglected.

Thus w e find the am to a sufficient approximation. The I functions can be found in tables, and the s6cond term in 3.1 (18) easily calculated. Unless [e is very near indeed to ,uo, the terms decrease rapidly, and we require only the first few terms.

3.4. The Distribution of Circulation.-From 3.1 (18), 3.2 (9) and 3.3 (8)

[k]fr G (~L) - 02mO(UOA 12, '2rn1 (2m + 1 V,), 10, 17R; = G (tt) _- E -2 2A -m Ie l(m + (1) 7r m _ +m I2m+i (2rn V1 , ) where G (,u) is the expression in curly brackets in 3.2 (9) and is tabulated in Table I, Am is given in 3.3 (7) and 4 in 3.3 (9), (10), (11) and the paragraph following.

As mentioned in the introduction, [0] is equal to the circulation round the corresponding blade section. If now we restore the factor wv/to, temporarily taken as 1 from the end of section 2 onwards, we have

__ 2 ___2 1______m____~L

9( AmS) (2) 7CWV m -=O 1+ 02o I2 ?1(2 + I o)

* This will increase the possible error in the calculated value of I'w/rwv at b = 3'8 when u, is 4 to about 7 or 8 in the last figure. Consequently, the corresponding point in the graph may be displaced this amount if the curve can thereby be smoothed. The actual displacement was from 0*334 to 0* 330.



450 S. GoldsteinL.

There is no further difficulty in computing the expression on the right. The values for = 2, 3, 4. ... 9, 10 are given in Table II below. These are the

Table I.-The Values of F7c4r7wv.

><\ 2. 3. 4. 7. 8. 9. 10. _ _ _ ~~~. 6. 7 TN. ...<.....___ii-

0.0 0 0 0 0 0 0 0 0 0 0-2 0*092 0*111 0*120 0*124 0*125 0-126 0*126 0*126 0-126- 0*4 0 *175 0 213 0 232 0*240 0*243 0 245 0*245 0*245 0-245, 0*6 0*243 0*303 0*331 0*344 0*349 0*351 0*352 0*352 0-352 0*8 0*295 0*379 0*418 0*434 0*441 0*443 0*445 0 445 0*445 1.0 0*329 0*440 0 489 0*511 0*,520 0 523 0*525 0 526 0*526 1*2 0-341 0*485 0*548 0*575 0*585 0*590 0*592 0 593 0*593 1*4 0*331 0 514 0-592 0 626 0 640 0-646 0 649 0 650 0 650 1*6 0*295 0 533 0 628 1 0-669 0 687 0-694 0Q696 0 698 0 698 1.8 0 220 0 537 0 654 *0*704 0 724 0,732 0-736 0*737 0-738 2-0 0 0 525 0-670 0-731 0 755 0-766 0-769 0^771 0-772 2 a5T 0.427 0 676 0 770 0*810 0 826 0 832 0 835 0 836- 2-8 0 303 * * * * * * 3 0 0 0-621 0*775 0-838 0 863 0 873 0 877 0*878 3.5 0*486 0-747 0 846 0Q884 0 899 0 905 0Q908 3'8 0 334 * . * * * *

4 0 0 0 671 0 830 0-890 0 915 0 924 0-927 4.5 I 0o519 0 786 0 882 0 920 0 935 0.940 4-8 0-351 * * * 5 0 0 0*701 0'858 0-918 0-941 0.950 5-5 0 543 * * * *

5-8 0-368 * * *

6.0 0 0-717 0 874 0-932 0-955- 65I 0 554 * * * 6 8 0*376 * * *

7 0 0 0-728 00883 0-941 7.5 0560 *

7*8 0*382 * *

8.0 I 0 0 734 0 890, 8-5 0-56 * 8-8 ! 1 36 9^0 I 0 0 738 9.5 I 0.569 9g8I 0388

10 0 1 0

* Not calculated.

calculated points from which the full line curves in fig. I were drawn. It may be mentioned that in terms of the usual parameter J, equal to v/nD, where n

is the number of revolutions per unit time and D is the diameter, we have

tlo ~7/J. (3) In view of recent and possible future developments, values of [to as low as 2 were considered.

In general, the error in any entry in Table II should not exceed 2 or 3 in the last figure.




In the notation here used, Prandtl's approximate solution is

F = 2 t2 cos-1 e-f (4) 17tvv -W. 1 +

where f (1 - j/j) (1 + [tIo2Y. (5)_

This is shown by the broken lines in fig. 1. 3.5. The Determination of the Flow.-The potential for the flow for r R,

0< 4 <vt, is given by 3.1 (17), where the a.m have been determined to a sufficient approximation. Restoring the factor wvl/o,) we may write, using 3.2 (8)

H 2 z O 2m+1 4_4S 2f+t(A2 cos (2m + 1) 4F WV -1 + [L 2M O (2m+1)2 7C _0(2m+1)2

00 tlo

)12m?1i(2m?li- L) _l+ ( ?L - -Zln cos (2m + 1)2

- =O v + .0 .2J+, (2m + 1 o)

+ IL2 7r - = (2m + 1) 1

m=O \l+4. J2m+l m+2)sin(l(2m2+I).()2

+ z ( A -&: )oI2m+, (2,+l 0) cos (2m + 1) (1)

_

o + ,Uo 12., 1 (2m? + I tio)

Since F2m+i ( ,u) is given in Table 1, this is easily calculated. We notice that the first series in the last term of 3.1 (17) cannot be differentiated term by t rin with respect to C, since the resulting series does not converge.

Also

a?, _ _ T'1, 2m+l c(s2 1 GO (2m + 1) WVL 7 = 2O m2m+1

z 0 v An E m) I2,+1 (2m 2m )(n+) n+ 1)C (2), - + [L2Am sm' I2?2mm1~ ( L1) s 9'n \

M=O\'1+1o2 12m+1(2m +IlO)

The first term approachest tr 2 3 )c-o) b Sintce

2sm,1 (2m + 1 O)/I2m?c+1 (2m + 1 i0)



452 S. Goldsteil.

_ _ _ _ , _=_ C

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4--

4-) a24.

4.4 _..1 Q-

I~ ~ !{ X--- , Rp I~~~~~~~~~~~~~~~~~~~~~~~~~~~D _4

-iTI- -- --7 V SQnR>t(Q

I I --- 1><S(x 0 4 z X X? z S i > M o

~~ ~ ~ ~ .41

cn C

- .Dr

0

P4 7~~~~~~~~~~~~

_ t_ .\ {7lI7 __ _ f o_ TXw|IslP P <r :'C?Q? O

ct

0 0 0




tends, with increasing m, to become independent of m, we can easily see from 3.3 (7) (p. 448) that the pecond term becomes infinite at t [L, like a multiple of (1 - p2/tL 2)-1.

If ur, uo and u, are the fluid velocities in the directions of r, 0 and z increasing respectively, then

?raL = (4)

6 L- (5) V

and

ut =Va, (6)

Hence the fluid velocities at any point are easily found from (2) and (3). At the surface, uo and u, are determined by the boundary condition alone,

and are We - WV0( + 02)~ fz _W tk1/(j + i2) (7)

At the propeller the velocities are half as much, if the contraction of the slip- stream be neglected.

The potential for the flow for r > R is given by 3.1 (1) (p. 443), from which- we see that the flow dies away rapidly as r increases. It is of some interest to find approximations for the constants a. This is done in Appendix 3, and the result is

~2 cm

where 1 3 1 .3. 5

=- =2-+ , 3 =246' (9)

Expressions for the fluid velocity can be written down at once. Thus

u w -W &g [ 2(2i V) cos 2+2 1* 4(4k1) cos 44 + ...], (10) + 022 K2 (2 l0) 2. 4 K4(41i0)

and so on. 4. The Solutionfor any Number of Blades.

4.1. Let us suppose there are p blades. Then

-!aaa)2+ (I + 02)+ =? 0(1)

aj

_~~ WV 27c 47C 2 (p7 )

and ~ L+ 2 eLforr?R, at =0, _ _ , (2)

and grad + vanishes for r = so . VOL. CXXIII.-A. 2 E



454 S. Goldstein.

The solution for r < R, 0 < ; < 2-v/p is

8 G Tljpt(m+j)(m?+ pj)co WlJV )P m-o (2m + 1)2 cos (2m + 1) 2

_ 2 v E A Ip(m?(m+ P.) cos (2m + 1) (3) p 1 + Lo2 m=o Ip(m+D) (m + p 2o)

where, approximately, the Am have the same values as before, namely,

Ao-1, 3A=- 5A = 1 .3 (4)

The exact values can be calculated in the same way as for two blades. The approximation becomes better and better the greater po and the greater the number of blades.

For r> IR, o4 _ 2 1, C,, Kr, (pn/) sin pn?, (5)

WV p 1 + {l02 -=I 2n Kp,,(pn io0)

where, approximately,

CIL == 21, C2 -2 . 4 C3= 2 * 4 * 56 > 6

As before, T131, (m+-) (m + 2- p CU) is near to [2/(1 + ,2). We put

2

+ - Ti,p +- (M + p ) = Fp, 2m+1 (t-), (7)

so that 0 Ti,pm+ (m + P-) TC2 2 o F F2nt+ m=O (2m+1)2 8 +u2 M-O (2m + 1)2 /

and we find for the distribution of circulation

ph1)r "2 8 XFp,2m?, ) 2itwv I + t2 72m=o (2m+ 1)2

2 , Amp(n+D(m + iP0) . (9) 7 1 + [Lo M=o I (M.t + (m + p ipo)

The Am are given in (4); the I functions are the modified Bessel functions, defined as in Watson's 'Bessel Functions.' The F are related to the T




functions by (7); to calculate the T functions we have formule 3.1 (14) and (15) (p. 444), and formula 3.2 (1) (p. 445). Also

Tl,p(m+t) (m + ipi) = o ([L) + '2(2 ) + 4 () + (10) +2(2m+1)2 p4(2m + 1)4 where

'ro = 02/(j + 12)1 (11)

and

Tr+2 (W_)-) (12)

so that the t are the same as in 3.2 (3) to (7) (p. 445). The only new point that arises for a three-bladed propeller is that the Bessel

functions 13/2, 19/2, ... are not tabulated.* There are simple finite expressions for these functions, and to calculate the value for any one value of the argument is not difficult. But to make even a rough table involves a not inconsiderable amount of work. The numerical work for the three-bladed propelleris therefore left over for the present.

4.2. The Four-bladed Propeller.-For a four-bladed propeller we have to calculate T1., (z), where v is an even integer. The analytical peculiarity mentioned in the footnote on p. 444 enters, the expression 3.1 (15) for T1, (z) becoming indeterminate. However, the asymptotic expaiision 3.2 (1) (p. 445) terminates, and is equal tot Si, (iz) or Ti,v (z) + (-l)iv vK, (z), so that T1,, (z) can be calculated from the formula

T1, (z)-= 1 _ V2/z2 + V2 (V2 -22)/Z4 V v2 (V2 - 22) (v2 42) z6 +

_ _)VvK, (z). (1)

The numerical calculations were carried through for o = 5, the approximate values of the A, being taken as in 4.1 (4) (p. 454). The values of 2 Po4rwvv, as given by 4.1 (9) with p equal to 4, are shown in the table below (Table III) and by the full line graph in fig. 2. The values of F2 1 (,u) or 2O/(l + 02) - T1,2 (2' ), are also exhibited. It was not necessary to calculate

F2,5 (pt), and the term in F2,3 (pt) never contributed more than 1 to the third figure.:

* No tables are recorded in Watson's 'Bessel Functions' or in Mises, 'Verzeichnis berechneter Funktionentafeln,' Berlin, 1928.

t Watson, ' Bessel Functions,' ? 10.71, p. 347. If the F are neglected, and we put ,x2/(1 + ,L2) for T in (1) with v equal to (4in + 2)

and z equal to (4m + 2) ,u, we have an approximate formula to calculate K. This method, for example, gives K6 (1 -2) equal to 1 197 *4221 instead of 1197 4227, as given by Watson.

2 H 2



456 S. Goldstein.

Table III.-Values of F2 1 (,) and 2rwP/7wv for a Four-bladed Propeller for

' F211 (p). 2 rwl/3WV. - - | , 1 () r 1

0.1 -0.0150 0*022 1*8 0-0157 0-750 0 2 -0 0341 0.066 2.0 0 0152 0-785 0-4' -0Q0517 0.180 2.5 0*0115 0-848 0-6 -0-0432 030- 3.0 0-0077 0-881 0.8 -00248 0*411 3.5 0.0050 0.887 1.0 -0*0075 0*506 4*0 0*0033 0*851 1*2 +0 0047 0*586 4-5 0*0021 0*714 1'4 +0*0117 0-652 4.8 0-0017 0.505 1.6 0.0149 0-706 5.0 0*0015 0

0.9~

06

0-7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- AI~~~

022 l~~~~~~ ~ ~ ~~-------- t

. I 3 4 UPr/& 5 Fia. 2.-THE DISTRIBUTION OF CIRCULATION ALONG A PROPELLER BLADE FOR A- FOUR-

BLADED AND FOR A TWO-BLADED PROPELLER, WHEN THE ENERGY LOST IN THE SLIP

STREAM IS A MINIMUM FOR A GIVEN THRUST.

The full line curves give the exact solution, the dotted curves the Prandtl approximation. In each case the curve with the greater maximum ordinate refers to a four-bladed propeller, the one with the less to a two-bladed propeller. The ordinates are pr(o/2irwv, and the abscissn wrlv, wherep is the number of blades and the other symbols have the same meaning as before (fig. 1)'. The curves are drawn for cR/v equal to 5. As before, each full line curve and the corresponding dotted curve are drawn for the same value of w.




The Prandtl approximation for a four-bladed propeller is 2rco 2 [2 cos' e2( 7CWV 7C + 0

where f = (1 + tLo2)4 (1 -,0LO). (3)

This is given by the broken line curve of fig. 2.

5. The Application of the Solution. Let

K _ (1) 2-t wv

where p is the nrumber of blades. We shall assume the distribution found in the previous sections, so that K is a given function of r/R for any given value of ,lo, or &R/v.

Let T be the thrust and Q the torque of the propeller, p the density of the fluid in which it is operat,ing, and CT and CQ the non-dimensional coefficients defined by

CT = T/7rpRv2, (2) and

CQ = COQ/_pR2v3. (3) If is the efficiency,

= vT/oQ = CT/CQ. (4) Let uo and uz, be the circumferential and axial components of velocity far behind the propeller, as given by 3.5 (7), so that uo = - w~x/(l + p2), and tf = w02/(1 + 02). The velocities at the blade are half as much if the con-

traction be neglected. Let

r/R =x and w/v = X. (5, 6) Then

y= wr/v- = x (7) Let us first neglect the profile drag of the blade sections. Then

=ppI (rw + 2uo) =ppr rw-iw 1 ) (8)

from which we find =2XkKx- -2K-

dx I + pa2x2 (9) so that

XT= 2?IJ - X212, (10) where

' Kx dx I,=3Kx dx, and I2 (11, 12)



458 S. Goldstein.

1, and '2 are functions of 7, most easily found by plotting the integrand and using a planimeter. They can be plotted against [Lo once and for all.

Further

dQ - ppPr (v + u?) -pprr(v+ < 2) (13) .

fromn which we find

dcQ - 2 ?Kx + 1,2 (14) dx l+[ (1

so that eQ _ 2?&I + X2 10213, (15)

where

o2J8 = 2J Kxdx + =1-I2 (16)

Hence

I - -f 1`o2A (17)

Thus, working at any given value of VO, we find X in terms of CT from (10), and then 4 from (17). We thus get a series of curves of 4 against oT for different values of Vu, or of ' against tuo for different values of CT. The thrust and torque grading curves are given by (9) and (14).

The theory holds only for small values of ?, for which we have approximately

1 ~ = - 1 (18)

1 + 02X I3/I1+12/1 1 +N211

and X CQ/211. (19)

We easily find that the energy lost per unit time is 2tpv3R2?n2I1.

If 0 is the blade angle, the incidence of any section is 0 - 0, wheret

tan v '2u, L1+ I I(+ y o2x2)=-L(1 (+U) approx. (20) tan = + lU = I - j/(l + 102X2) f @ + -2fUB 'ILO; 1 2 [L1+V OX

We pass on to consider the effect of profile drag. Let s be the ratio of drag to lift for a wing of infinite aspect ratio with the same section as the blade element at a distance r from the axis.t The contribution of the drag to the

* This form is preferable to 1 - IA,, since eo2la is greater than '2,

-t The symbol 0 is differently used in sections 2, 3 and 4. The symbol e is differently used in sections 2 and 3.




element of thrust is then - e dQo/r, where dQo is the element of torque in the absence of profile drag. The contribution to the element of torque is similarly sr dTo, where dTo is the element of thrust in the absence of profile drag. Hence

dcT- d21 _ , d) dx dx pLox dx

and dxQ dxQ dc^ (22) dx dx+ dx

dcrjdx and deQ./dx are given by (9) and (14). Hence dcT 2Kx 2 Kx K2X _ 2 2 (23) dx 1+- i2x2 fLo I? + Fx' (3

and

dQ= 2?Kdx+ 2Vo2 Kx+ 3 +_2X2 (24) dx 1+~222Lex-~ + 1024

Consequently GT- 2J1 - X21 2 4 X2 .L05 (25)

and cQ= 2X1X + X2 v0213 +2X[016 -X2oI5, (26)

where 14 = eK dx, (27)

0

I Kx2dx (28) J1 1 ?(28x2

and I6 eKX2 dX. (29)

Then CT - l - Jd2-I41/0o-PLoI5 30

4 Jc + I 1+W -Xuo2Is+ pL0I6 - 7LoJ (30)

For small values of X and s,

I+ -1X+ 17/11' where

17 4IUVO ?+ {toJ6 sK (1 + to2x2) dx. (32)

Thus the influence of friction is of order e0, and friction has more influence on propellers the greater the ratio of tip speed to forward velocity. The



460 S. Goldsteini.

determination of e depends upon the incidence, and therefore on the angle f. For some purposes it is advantageous to take a rough average of s, and write

i I7 K K( + ,t2x2)dx. (33) Pto o

The integral is then a pure function of Ao. Any losses dependent on the varia- tion of the distribution of P from that taken here can be thrown into the S term.

For accurate work we take a set of values of X, calculate s from (20), find e and determine 14 15 and 16 graphically. The corresponding values of cT are given by (25), and of by (30). Hence we get the curve of n against CT for the given blade and the value of p0 considered. We can get a set of curves for different values of pt0, which can be turned into a set of curves of n against pt for different values of cT.

The above considerations are valid only if ?, or cT/21i, is small. For moderate loads we get a better approximation if, following Prandtl, we replace v by v + 9gw. Again, according to Betz, 'Handbuch der Physik,' vol. 7, pp. 256- 259 (1927), the contraction in the slip stream can be allowed for by considering a screw surface for which V.0 is coR/(v + w), so that the solution of the potential problem given here will still retain its usefulness. But there are still difficulties to be overcome in applying it, and a closer discussion must remain over.

My thanks are due to Prof. Betz, who suggested the problem to me.

Summary. The distribution of circulation along a propeller blade when, for a given

thrust, the energy lost in the slip-stream is a minimum, is calculated exactly and compared with the approximate Prandtl formula. Numerical values are given, for a two-bladed propeller, for various values of coR/v, where co is the angular velocity, R the radius, and v the velocity of advance of the propeller. From these values the curves in fig. I were drawn. An example (coR/v 5) was also computed numerically for a four-bladed propeller, and the result is shown in fig. 2. The distribution for any number of blades and any value of 6R/v can be worked out numerically by the method used, the work being easier the greater the number of blades and the greater woR/v. Formulie for the fluid velocities far behind the propeller were found, from wbich numerical values can be worked out by the help of methods and numerical tables given.




Formulae are set out for the distribution of thrust and torque, and for the efficiency, when the distribution of circulation is that calculated.

APPENDIX 1.

The Solution in Trigonometrical Series for a Rotating Lamina.

If the screw surface moves along its axis with velocity w and rotates with angular velocity cowlv about its axis, its displacement is merely along its own surface. The fluid motion is therefore the same whether the surface moves along its axis with velocity w, or rotates about its axis with angular velocity cot, equal to -cow/v. If co/v becomes zero while co, remains finite, we fall back on the case of the rotating lamina, whose solution* is

+ i+ .1- iR2cO.e-2t, (1)

where t = i + iii, and i, n are elliptical co-ordinates given by X-==retg = R cosh t.(2

The solution is easily put into the form of trigonometrical series. For

Re-t - - (,2 R2) (3) Hence R2e-2t _ 2X2 - 12 - 2-7 (X2 - R2)1

= T 2 F2R 1- R2 2-4TRh2.4.611 * ' -~~~~~~~2 I r4 I I 38 fr 6

2r2 -R2 2,i I - 1R 1.1R4 _ 1.1.3 R -2 2 R22 .4 d 2 .4. 6 R J

for1 tJ >f R. (4) Ilence

- ~ ~~~ 72 .4.6()CS - f} for r 6R

~~~~~~~~~~~~~ 4 o56 'COS co

T -hRe sins2on+tiuiyin +ifcrr(2 - 6> )R. (5) Trhe discontiniiity in 0 iS o)lr (R2 -r2)-1.

* Lamb, ' Hydrodynamics,' chap. iv, ? 72.



462 S. Goldstein.

APPENDIX 2.

The Solution of the Potential Problem for .R/v small.

We return to the potential problem formulated in section 2, namely,

(a2 a2cf I V.07a 2 + (I + ,U2) 0)= > 1

=-1 at42 rat for 0<fOr <r AR, (2)

grad b = 0 at r oo, and b is a single-valued function. We assume

=I a.12,, (2n[t) sin 2nr + E bi n2m+1 (2C + 1 - cos (2m + 1) 4 ni=l m=O 1I2m+i(2m+IiLo)

for r<6 X O< 4< 7x (3} and

Sb= EcT K2b (2nw) sin 2n? forr > R. (4)

Then

) = 2na,12,,(2nt). (5) ()=O or ,r n=1

Now

,?___ - 2 3 1)"' '2m (2ny), (6)

if , is less than 0*66 ... (Watson's 'Bessel Functions,' Chap. 17). Hence, if Vo < 0.66 ..., we can take

a, = (-1)hn. (7)

The constants bmn and Con are then to be found from the conditions of continuity at t = 0, namely, the continuity of S and of a S/ar. This can be done numerically as in 3.3, by expanding cos (2m + 1)' in a sine-series; or by assunming all but a finite number of the bm and c,, to be zero, and equating the expressions for Sb, and also for aSlar, at a finite number of selected values of 4 between 0 and n. The convergence will be slow.




APPENDIX 3.

The Approximate Determination of the Constants am and c". 1. Let

ct0 '2n 00 2mn + = 2nI (ra! sin 2n0 - O (ra,r cos (2m, + 1) 0

for r < a, 0 6 0 < 7

z B(%) ?sin 2nO for r a. (1) r

For r < a, -7X < 0 < 0, let b be as in the first line with the sign of the second term changed.

Expand cos (2m. + 1) 0 in a sine-series for 0 < 0 < r, and equate coefficients of sin 2nO. We obtain

1 4 ~~2n A Bn-2n 7 m7 O 4n2- (2m+ 1)2 (2)

Differentiate the two expressions for b with respect to r and repeat the process. This gives

-B-1 o 2m + A - 2n c 7-E0 4n2 - (2mki + 1)2 IT (3)

Eliminate B.. Then X, Am _ E m - = 0. ~~~~~~~~~~(4) m=o 2n -2m- I 01

The equations (2), (3) and (4) hold for all positive integral values of n. The value of o so found is the solution of the problem defined by

a2?> a2X

?~~ ~~ - 0~(6

((~#) = i ('\ 2

2r rao O=0orr rn=l a a-r

grad b 0 at r oo, b is a single-valued function. This problem can be solved in finite terms by means of elliptic co-ordinates i for which

t _ Z + i_, (7)

z -rew a cosh t. (8)



464 S. Goldstein.

The boundary condition becomes

a=cot at i =0, (9)

and the solution is

+ i= i log (1-e2). (10) Now

-e-21 2z2a2 - 1- 2(z/a) (z2/a2 -1)-, (11) so that

Ok+i4 z=ilog2+ilog[1 -zs/a2 +(z/a) (Z2/a2 1)I] = log 2 + ,li log (1 -z2/a2)-sin-1 z/a = i log 2 - ji (z2/a2 + z4/2a4 -- z6/3a6 + )

(z I.1zW3 1.31z5 1. 3. 51z7 ( 2 .7 + ~ ~ ~~12 'a 2.3a3 2.45 5+24.67a7+ 24( Thus

=)2 sin,20 + 4sin 40+.)

(~~ 0+~~ r3 cos30 3L - cos -a cos 0 + 2. 3 a3 - 30 + 2.4 5a5 cos5'+ (13)

and the solution of the equation (4) is

Ao = 1, 3A1 = - 5A 4' 7A3 1. 3. 5 (14) -2; 2 2.4' ~2. 4. 6 The solution of the equations 3.3 (6) is then

2

a.= + l A (15)

where the A, are given by (14). 2. With the approximations described in 3.3, the equations 3.3 (3), p. 447,

become 4 00 (2m +1) a. (6 7. 4n2 - (2m+ 1)2

This is the condition that 00 00

E (2m + 1) am cos (2m + 1) 0=- 2n c, sin 2nO (17) =m0 n-1

for 0 < 0 < 7.*

* This comes directly if we differentiate 3.1 (1) and 3.1 (17) term by term with respect to fi, equate the results for [z equal to go, and make the approximations described in 3.3.




Omit temporarily the factor - 02/(1 + 2) and take

(2m + 1) am,l_,1 . 3 . 5 .(2..2 ) (18) Let

a0= - (2m + 1) acos(2m + 1) 0, (19) =OQ and

2- (2m+ 1)amsin(2m +1)0, (20) m=o so that

0+i

-A jl ?1 2 ?Le-2z + 4 ?jI (21)

Then 3 ~~~~~~~~(22) =sln-2 2.0 + /

0 + and

$==+(1+ cos+20 + cos40+... ). (23)

Putting X jn in S, we see that the upper sign must be taken. Restoring the factr - A2(!( + o2) we have finally

ff I F s2n (24) I + 2 where

=2=23 - 3 1 .3 5



On the Vortex Theory of Screw Propellers

Documents

Transcript of On the Vortex Theory of Screw Propellers