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Resistance to a Barrier in the Shape of an Arc of a Circle Author(s): L. Rosenhead and S. Brodetsky Source: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 117, No. 777 (Jan. 2, 1928), pp. 417-433 Published by: The Royal Society Stable URL: http://www.jstor.org/stable/94857 . Accessed: 04/05/2014 16:57 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 of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of 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 the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. http://www.jstor.org This content downloaded from 62.122.78.35 on Sun, 4 May 2014 16:57:58 PM All use subject to JSTOR Terms and Conditions

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Page 1: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistance to a Barrier in the Shape of an Arc of a CircleAuthor(s): L. Rosenhead and S. BrodetskySource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 117, No. 777 (Jan. 2, 1928), pp. 417-433Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/94857 .

Accessed: 04/05/2014 16:57

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

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Page 2: Resistance to a Barrier in the Shape of an Arc of a Circle

417

Resiet ance to ab Ba6rrter in thc Shape of an Arc of a Circle. By L. ROSENHEAD, B.Sc.

With Note by S. BRODETSKY, M.A., Ph.D.

(Colmunwicated by L. Bairstow, F.R.S.-Received July 7,1927.)

Comparatively little progress in the solution of the problem of discontinuous fluid motion past a curved barrier was made until Levi-Civita formulated a method of transforming that part of the barrier which is in contact with the moving fluid into a semi-circle in an Argand diagram. This, indeed, was the starting point of much work of interest and importance. Useful accounts of the problem of motion past any barrier, together with extensions, are given by Cisotti* and Brillouin.t Leaving out of accouLnt such barriers as are made up of one or more planes, problems which can be solved by the older methods based on Schwartz-Christoffel transformations, the only applications of Levi-Civita's method to curved barriers seem to be that made by Brillouin in the paper referred to, and those made by S. Brodetsky in 1922.' The work of Brillouin, however, and that of other investigators? are essentially backward processes, in which a likely expression is written down and the streaming motion implied, as well as the shape of the boundary, are investigated. A more direct attack is obtained by suitably choosing the coefficients in Levi-Civita's general formula, and arriving at the solution for a given curved barrier by a series of steps in s-uccessive approximation. The solation of the problem for a circular barrier placed symmetrically in the streaming fluid has been obtained in this manner by S. Brodetsky.l

The object of this paper is to solve the problem of the circular barrier placed in any position in the streaming fluid, subject to the condition, however, that neither of the ends of the barrier are in the, "dead " fluid-i.e., the radius of curvature of the free stream line is zero at each end. This immediately restricts the barrier, if convex to the streaming fluid, to be of angtular extent less than 110 * 20.?

* 'Idromeccanica Piana,' vol. 2, Milano (1922). t 'Ann. Chim. Phys.,' vol. 23, pp. 145-230 (1911). ,j: 'Roy. Soc. Proc.,' A, vol. 102, pp. 361 and 542. ? See, e.g., Creenhil's " Theory of a Stream Line past a Curved Wing," 'Advisory

Committee for Aeronautics,' Londoni, 1916. See " Fluid Motion Past Circular Barriers," ' Scripta Univ. Bibl. Heirosolym.,' vol. 1,

No. xi (1923). ? Ibid., p. 8.

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Page 3: Resistance to a Barrier in the Shape of an Arc of a Circle

4138 L. Rosenhead.

A circular barrier can be either concave to the streaming fluid or convex to it. Using the ordinary conventions in the z plane of fig. 1, the angle of contingence at A measured from C is positive for concave or positive camxber, negative for convex or negative camber. The kind and angular extent of the circular barrier will therefore be indicated by the sign and value of the angle of contin- gence at A, measured from C. It has been proved in the last-named paper that there is an upper limit to the arithmetical value of the negative angle of contingence at A--namely, about 55- 1--for otherwise the free stream line would leave the barrier at some point between C and A.

The motion is two dimensional. Let the complex variable z (-x + iy) define position in any plane perpendicular to the generators of the barrier, the x axis being parallel to the direction of the stream at infinity. We define

,b - = V ax ay _y ap

where u, v are the velocity components, and +b, + are the velocity potential and stream function, respectively. Let w _q + i4 and define C, 52, r, 0 so that

=retOdz-; Q =log =logr+i0.

Fig. 1 shows the z, w, 4, Q , T planes for such a problem. C is the point of bifurcation of the stream line IC; CA, CA' are the stream lines in contact with the barrier; AJ, A'J' are the free stream lines. We take the standard dimen- sions to be unit velocity at infinity, unit value of w (\/CA + \/CA') as measured in the w plane, and unit density.*

We introduce the transformation in which the variable _- pel? is given by

/w= 2( -- -) sin ao, where the point C is given by -= -eL. It is

immediately verified that the barrier ACA is the semi-circle p =1,-2 < _

2 2 in the X plane, while the free stream lines AJ, A'J' become the two halves of the diameter along the imaginary axis, namely, L = - 1 ? p ? 1. In the general problem we write

X log {(1 + el76.)/(1 - e 1o-) + Al-c t- LA2 2 + IA3t3 +-

- a1-~ +X~a2 + Aa3- 8 + a44 + ..., (1)

where the A's, a's are all real and the expansion in (1) is convergent for I X I < 1. ?7 is the angle at C measured away from the streaming fluid.

* See Brodetsky, 'Roy. Soc. Proc.,' A, vol. 102, p. 361 (1922).

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Page 4: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistance to Circular Barrier. 419

C

A A

(AA A A'

C R # ~~~~~~C

A~~

A C-_t Log OT

A'

C

A'

FIG. 1.

In the case of the circular barrier, the angle between the two tan( gents at anly two points on the barrier is -x, hence X = 1. In addition we have

a1 = A1 + 2 cos a0 a2 A2-2 sin 20,...

a3 =A3+ 2 cos 3a0, a4 A4- 2 sin 4c0, etc. On the barrier, t = e7; therefore

Q = log {(I + eT+?TO)/(l- tCLf?o)} + A1eT -+ YLA2 e2lt + I-A3 e3t ..".

logr+ LO.

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Page 5: Resistance to a Barrier in the Shape of an Arc of a Circle

420 L. Revenhead.

We get

log r -log + F A1 cos a- ! sin 2cs

3 !A3 cos 36 - 4A4 sin 4a..., (2)

0 tan1 sin (Cy + jIo) tan--1 "in (6 6

+ COS (or + C0) i-eCos (a --0)

+ A1 sin c + IA2 cos 2%o --A3 Sill

?i -2I1- go si Al sn o + WA2cos 2a + --A:3 sin 3a.... (3)

(We use ? /2 according as 0 ao.) The angular extenit t of the barrier

(OA _-

OA') -_7

= 2A- 3-A3 k 1-S -47 ...)*(

Now r -- I dz/dw . On the barrier l dz ds, and w = 54. Therefo:re

r - ds/db.

Also \/ (sin a - sin ao), therefore

ds/da - r (db/d) - 2r cos a (sin a - sin 0o).

If R is the radius of curvature of the barrier,

ds ds/da dO dO/da

o 1 -- cos (a +a El 1(A, cosa - 1-AA2 sin 2o+ -A3 cos 3...) s o - cos ( -- a0)J Al cos a -- A2 sin 2a + A3 cos 3a...

2{l +COS (G + 60)}E (A, cos r - A2sin 2a + 1A,3 cos 3a ... (

A1- A2sin2a +A3 cos3a cos a cos a

T'lie A's maust now be chosen in such a way that R Is constant, i.e., independent of a, between ? n /2.

Consider the free stream lines. Along AJ we must put rtp, 0 < p ? 't. Therefore,

r 1, and 0 -alP -2a2P _- 4a3p + ja4p + -1- 5p5 ...

The stream line AJ must be conlvex to the movinig fluid-i.e., ds/dO must be negative,

ds --?q d 4rd {- d (p -- l/p) - silloj2dp/dO.

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Page 6: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistanme, to Circular B rrier. 421

Therefore

ds ds dp ds df dp dZ dp dO d(I Ydp dO

d- dp ( p2) 1- (p + I/p)- sin 8 0}2 dp dp'dTO p dO'

Therefore dO/dp must be positive in the range 0 ? p ? 1,

al - 2 P 3_ 3P2+ 4p3 + a5...>0 0?P?1. (6) Similarly along A'J, by putting X - ip where 0 ? p ? 1, the condition

that the stream linie is convex to the flaid is found to be

+ 2P-3P2 -a4p3 + a5p ...> , >07 0 p . (7)

The angular extent of the barrier, and the angle inade by the axis of the barrier with the direction of streaming, completely specify the conditions of the problem. The results will depend on these two quantities, and since the quantities are entirely independeit, we should expect the final result to be a function of two param eters.

As a first approximation pult A4 A= A6 ... 0, so that we can arrange that the radius of curvature should be the same at three points on the barrier. The points chosen are a =7r/2, 0, - /2, and so we postulate that

2(1 -s) E (Al+-IA3) _ 2(1+s) A1 - 2A2-3A3 Al + A3 A, + 2A2-3A3'

where s sin ao and c cos c0, and each fraction represents the radius R. Choose as our two parameters

Go and (A, -4 A)

Hence

3 (6O: +I log 1e 2ii 8 Al - 3(+)1 2 Y ' 2 (10( + 1) oge 1 ;)

2 (100 + 1) (1--VcJ

A3 - (~ilog 2' ~2(10~ +i) e\1 /1c

We note now that i must be positive, for 2= (1 + c) eAl+IA' which is always positive. Also the angular extent is

2 (A- 3A3) - jj) log L (9)

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Page 7: Resistance to a Barrier in the Shape of an Arc of a Circle

422 L. Rosenhead.

Hence if the barrier is convex to the stream, f must be negative, i.e., 24 < (1 + c); if plane, A1A2 A3 = 0, iex., 2 = (1 + c); if concave 2S > (1 + c). We note also that when , is negative, A1 and A2 are also negative.

R can now be expressed in terms of a, a. and i, and if it be plotted against a, having given definite values to a0 and i, the extent of the variation of the function R from its value at the three points a ==/2, 0,- 7/2, canl at once be seen.

It is now necessary to ascertain the possible values of i and a0, and any relations that may exist between them. These are found by considering the conditions that the free stream lines are convex to the streaming fluid.

The general condition that AJ is convex to the streaming fluid is

l-2P-3p2 +a4p + cp ... > 0

if in this we put A4- A5 =A .. 0, the condition becomaes

A1-A2p- A3p2+ 2c/(l-2ps +p2) > 0, 0 ? p ?1. (10)

Similarly, the condition that A'J' is convex to the streaming fluid becomes AI + A2p-Aap2+ 2c/(1 + 2ps +p2) > 0, 0 p < 1. ()

Go is taken positive, for the relative conf-iguration of the fluid and barrier is the same whether c0 is positive or negative. "If the barrier is convex to the streaming fluid, and if a0 is positive, then the axis of the barrier will make a positive angle with the direction of streaming, and the stream line will be in a more critical posi.tion at A' than at A. It is therefore conidition (11) that has to be considered. This can be shown mathematically as follows. Let

XI-A1-A2p-A3p2+ 2c/(l -2ps +p2), and

X- A1 + A2p - A3p2 + 2c/(l + 2ps + p2).

It is required to show that if condition (11) is satisfied, then condition (10) is satisfied automatically, i.e., X1-X2> 0, or

4sc/(1 + 2p2 cos 2ao + p4) > A2

This is true, for the left-hand side of the inequality is always positive, whereas A2 is always negative when the barrier is convex to the streaming fluid. Now let

f (p) A1 + A2p -,A3p2 + 2c/(l + 2ps + p2);

then f (1) Al + A2- A3 + C/( 4).

\Ve canl show that if f (1) ? 0, then f (p) > f (l) for all values of p. The

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Page 8: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistance to Circular Barrier. 423

condition that f (p) should diminish gradually as p increases from 0 -- 1 is that d{f (p)}/dp be negative for all values of p in the range. This gives

A2 -2pA3- 4c(s + p)/(l + 2ps + p2)2 < 0, i.e.,

4c (1 + (s/p))/(l + 2ps + p2)2 - A2/p> - 2A3. The left-hand side of this inequality is positive, for A2 is negative. Also the left-hand side diminishes as p increases, so that if the inequality is truie for p 1, it is true for all values of p. Hence we need

c/(l + s) > A2 - 2A3.

But we know that f (1) ? 0, i.e., c/(1 + s) > A3- A2- A1.

Also the difference (A3 -A2- A1) - (A2- 2A3) 3A3- 2A2- Al

= 6 (I + s) loge 2 ) 10~ + 1 +

and this is positive, since 20 < (1 + c). Thus (A3- A2- A1)> A2- 2A3

or c/(l + s) > A2- 2A3,

which is the required condition. Hence if A1+ A2- A3 + cl -+ s) 2 O, condition X2> 0 is satisfied for every point on the barrier.

In the limiting case-i.e., when the curvature of the free stream at A1 just becomes finite,

A1+A2-A3+c/(1 +s) = O, i.e.,

3 (2, + s?1) loge( 2+ )+ c 0. (12) 10 +l 1 I+ c 1+ s

Below are tabulated values of a0 and i which satisfy this equation. .~~~~~~~~~~~~,

degrees. Al. A2. A3. radians.

0 0 3970 - 0*9426 0.0000 0 0574 - 1 9234 9 0.4609 - 0 7738 - 00643 00161 - 1-5584

18 0.5159 - 06354 - 00959 - 00049 - 1-2676 27 0 5600 0 5190 - 0.1081 - 0-0143 - 1-0285 36 0.5918 - 0 4182 - 0-1080 - 0.0169 - 0 8252 45 0.6106 - 0*3299 - 0.1000 - 00156 - 06494 54 0.6152 - 0 2509 - 0 0865 - 0-0123 - 0-4936 63 0.6060 - 01793 - 0 9689 - 0,0082 - 0 3532 72 0.5827 - 0-1148 - 0*0486 - 0 0042 - 0-2268 81 0 5471 - 0.0549 - 0.0253 - 0.0012 - 0.1090 90 05000 0.0000 0.0000 00000 00000

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Page 9: Resistance to a Barrier in the Shape of an Arc of a Circle

424 L. Rosenhead.

The preceding table gives 4 and a0 corresponding to the limiting positions in which barriers of various angular extents can be placed. Thus, if we are given the general problem of a barrier of angular extent ( (which must lie between 0 and - 1I 9234 radiaus), it is possible to obtain the values of i and ao, corre- sponding to its limiting position, by i-nterpolatini, or byr solving for > andc; the two eqnaiftions

3(2E,-4](4 4) 721 +2 t,lle.~ ~ ~ ~~~~(0 1;)( eqI t3f,i?-

(2 I ( s) loge 9 IA( s (0 I- 1) 8 (3-k <:) 1 -d- s

can be eliminated from these equiations, giviLng rise to a single equation in a0 :^-

4c+ - P( 1?s) (4+ 5s) - 4 (I + s) (I + 2s)log [-(3 ((1+ S)2 + 4e)!(l + c) {3 (I + s) - 8c}l. (13)

In the general case, when the stream line starts with infinite cuLrvature at the terminations of the barrier, S and aO can have any value subject to the conditions that i is greater, and a0 less, than those values which correspond to the limiting position of the barrier under consideration. An upper limit for ao and a lower limit for 0 are thus determined. The following diagram makes this clear.

L 0 13= 0

__~Ge~ Limninosition .6 csCu-rve of

.,-1-9234radians

.2

0 IS 36 5- 72 90

Fie. 2.

The upper limit for corresponds to the symmetrical case-i.e., cao 0-and its value for a barrier of given angular extent is obtained from the equation

The case of a barrier convex to the stream is thus fully discussed. We can also reasonably adopt a limiting position for a barrier concave to the

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Page 10: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistance to Circular Blarrier. 425

stream, namely, when the tangent at either A or A' is parallel to the direction of streaming. This immediately restricts the barrier to be of angular extent less than 1800. We shall take co positive, so that the limiting case occurs at A.

In the general position we must have 7r> 0A, I.e.,

2 (100 -- 1) or

7/2- ao > 2 {14(34 ' }

The values of i and ao which make the tangent at A parallel to the direction of streaming are obtained from the two equations

7/2-rn?-2 {1 4(4 +1)}

45_ (44 + 11) loge (2 +C

ca n be eliminated from these equations, leaving a single equation in ao:- 53s-4( +2ao- ) 0og 33s 4 (? + 2?o-)

8s 8(I + c) (3 + 2ao-7) We have tabulated the upper and lower limits to i and ao for various barriers-

i.e., the values of the parameters corresponding to the symmetrical and limiting positions. The limiting values to i and a0, for convex and concave barriers, are separated by a horizontal line to indicate that they were obtained from difterent considerations.

e for efrC' o p p js \ symmetrical fiiiting i riting

in degrees. in radians. case. positiong iposition ao == 0 ~~~~~Degrees.

- 110.2 - 1*9234 0*3970 03970 0 00 - 100 - 1*7453 0-4277 0*4272 4 18 - 90 - 1*5708 0-4607 0*4586 8 39 - 80 - 1*3963 0-4978 04913 13 49 - 70 - 1 2217 0*5388 05247 19 34 - 60 - 1 0472 0-5847 0 5586 26 10 - 50 - 0-8727 0X6357 0-5853 33 46 - 40 - 0-6981 06928 0*6066 42 22 - 30 - 0-5236 0-7569 06158 52 7 - 20 - 0X3491 0.8287 0-6055 63 16 - 10 - 0-1745 0-9093 0*5683 75 58

0 0-0000 1*0000 05000 90 00

10 0-1745 1 1018 0-5837 86 7 20 0-3491 1*2163 0-6842 81 59 30 0.5236 1-3451 0 8041 77 36 40 0*6981 1*4899 O09464 73 00 50 0*8727 1'6527 1*1142 68 12 60 1'0472 1'8357 1'3102 63 13

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Page 11: Resistance to a Barrier in the Shape of an Arc of a Circle

426 L. Rosenhead.

Particular cases can be considered. To explain the method, one of these cases will be discussed fully.

Take the case f3 + 100, so that the barrier has angular extent 100, and is concave to the stream. Choose c0 = 86? 7', say, then the parameter i which satisfies the equation (9) is equal to 0- 5837. (As a matter of fact, these two parameters give the limiting position of the barrier.) From these values of i and ao we find A1, A2, and A3 from (8). R and 0 are then given for different values of a, ranging between + 90? by (5). Plot R, R sin 0, R cos 0, against

0. The value of f R dO (i.e., the length of the barrier) can be obtained by

graphical integration. Similarly the values of fIR sin 0 dO and f R cos 0 dO

can be found for various points on the curve. Since x fIR cos O dO and

y JR sin 0 dO, the Cartesian co-ordinates of any point on the barrier are

known. There is actually a big variation in R, yet on plotting, the barrier is found to be very approximately circular. The effect of the big variation in R. is, in fact, nullified by the very small range of 0 in which it occurs. The radius of the barrier so plotted is not, however, exactly equal to the calculated value of R at the three specified points. A second approximation to the value of R is

obtained by dividing the length R dO, obtained by graphical methods, by

the angular extent of the barrier under consideration. If an arc of a circle is drawn with this radius, the end points coinciding with the ends of the plotted barrier, the two curves are found to be almost identical. This method gives considerable accuracy in the case of small angular extent, and it is only when the angular extent becomes large-e.g., P _ 90'-that the error begins to become perceptible. Even then the error in the radius is at most 1 * 5 per cent.

In this way we can work out and tabulate pairs of values of values of 4 and a0, which, if substituted in the Levi-Civita formula, give circular barriers of known radii, angular extents and positions.

It is more usual and convenient to describe a barrier by its camber and angle of attack. We define the camber as

Sagitta of Barrier 1 Chord of Barrier = 2

Also the line AA' is parallel to the bisector of the ext-erior angle between the tangents at A and A'. Hence

1 (OA + +OA,) = V, i.e., o=f ( - 2o + A). (15)

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Page 12: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistance to Ciricular Barrier. 427

In our notation, the components of thrust on unit length of the barrier (measured perpendicularly to the z plane) are P,E' and P,', given by the equations

/-1 7:a 2 P' I7pt(a4-asn Px 4T 1' +4a, sin ao). (See Cisotti, p. 175, and Brillouin, p. 195.) if Px' and P,' are divided by 2R sin P/2, the resulting values are the componenits of drag and lift on a barrier of unit chord, but of the same camber and orientation. Let these quantities be P, and Py, and their resultant p V )2 ? p2

This resultant thrust passes through the cenitre of the circle, for the pressure is everywhere normal to the surface. Its line of action will cut the chord of the barrier in the point D given by

__ _ sinD sin tan( ' 1"+x- AD sin DOA 2 p 2J DA' sin DOA' sin + (tanip-' ?+ - j)}

Cos a 3octan-' ply 2_PX (1-6) CosHfP_Ll y~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2y i 2 m tan-' -

o Px~~~~~~~~~~~

0 PX FIG. 3.

This is valid for barriers convex or concave to the stream.

If D = 0, so that the barrier is flat, this ratio becomes - cos ( 7t/2) and is cos (-,r/2) therefore apparently indeterminate. In the case of the plane barrier, however, 2tY t( + c). Let us therefore put 24 = (1 + c) (1 + h) where powers of h higher than the first can be neglected. It can be easily verified that ( AD_\ 2(4 + c) -3s

h -J 0 DA'I 2(4+ -c)+3s' VOL. CXVII.-A. 2 G

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Page 13: Resistance to a Barrier in the Shape of an Arc of a Circle

428 L. Rosenhead.

and the distance of D from the mid-point of the barrier is

31s/(4 + s-i) (17)

where 1 is the length of the chord. The exact value of this distance is

31s/(4 + -rc).

The error divided by the chord is therefore

3(7- - -) sin 2ayo 1 sin 2co (4 + rc) (4 -c) 90 1 + 1 - 452 cos ao + O * 524 cos2 (18)

The maximum value of this expression is about ; so that we get an accuracy of about W per cent. by meanis of this approximation.

In order to demonstrate clearly the effect of the camber on the resultant resistance P experienced by the barrier, we give the resistance experienced by a plane barrier of unit length, but with the same angle of attack as the circular barrier. These figures are put in brackets above the figures giving the thrusts on the corresponding circular barriers. As is to be expected, the thrust on a concave barrier is greater, and that on a conrvex barrier less, than the thrust on a corresponding plane barrier.

Returning to the case of the circular barrier, it is possible to determine i and ao so that the barrier has given camber and angle of attack. They are determined from the following equations:

4 4 )loge (2~ (10- + 1)

oc=(rnI2) -ao? +og 2 (10 + 1) I + C)

can here be eliminated, giving rise to an equation in 60:-

2(1 + c) = 3es _ l] e- 8+ sin - . (19)

L8 {aO + oc -n/ i

In the table (pp.1430, 431) we give the angular distance of the centre of pressure from the centre of the barrier. It is

e tan-, PV + oc -T/2. Px

We also give the exact position of the centre of pressure on the chord D by

MD _ 1- AD/DA' AX 1 + AD/DA"

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Page 14: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistance to Circular Barrier. 429

(MD/AA') and P have been plotted against oc, and these graphs illustrate well-known phenomena associated with cambered aerofoils.

=4G 00 20= 00

\i>X X

+~~~~~~~~~~~~~~~~~~~~~/+

0 5 2 F1 X =31n \\ 1

T ,

1+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1

_ T - - _ *120 46 90'

Consider the graph of (MD/AA') against oc. In a symmetrical position the centre of pressure is, of course, at the mid-point M. As oc decreases 'we see that for certain barriers the centre of pressure moves up towards A. It attains a certain position at which its distance from M is a maximum, and then, as oc decreases still further, it returns towards M. This is very marked for small camber. A well-known phenomenon is thus explained mathematically.

:For barriers of comparatively large angular ex:tent, e.g., goo90, the centre 2 G 2

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Page 15: Resistance to a Barrier in the Shape of an Arc of a Circle

430 L. Rosenihead.

{.1 O. Al.1 al. A2 j 2- i

0 1 0.

0 5683 75 58 - 0-0881 0-3969 0-0388 - 0-9798 - 0-0027 25 39 - 10

0-6055 63 16 - 0-1772 0-7226 - 0-0683 - 1-6753 - 0-0081 13-58 20

0-6158 52 7 - 0-2661 0-9621 - 00895 - 2-0281 - 0-0131 9-22 30

0-6800 37 50 - 0-2681 1-3115 - 0-0648 -- 2-0026 - 0-0190 9-91 30

0 5837 86 7 0-0883 0-2237 0-0392 - 0-2312 0-0033 26-10 10

0-6842 81 59 0-1789 0-4579 0-0694 - 0-4830 0-0129 14-82 20

0*8637 64 31 0 1817 1 0422 0 0531 - 15005 0 0214 1784 20

0-8933 61 26 0-1820 1-1384 0-0503 -- 1-6296 0-0225 18-33 20

1-0649 40 56 0-1839 1-6948 0-0326 -- 1-9473 0-0281 21 26 20

1*1772 20 28 0-1849 2-0587 0-0161 - 1 -2943 0-0311 2348 20

1-2163 0 0-1852 2-1852 0 0 0-0320 23-88 20

0.8041 77 36 0*2712 0-7006 0-0910 - 0-7480 0*0283 11-17 30

0*9464 73 0 0*3653 0-9501 0-1046 - 1-0138 0-0488 9 41 40

1.0761 62 31 0-3681 1-2911 0-0876 - 1-5501 0-0569 10 70 40

1-1938 52 3 0-3700 1-5999 0 0715 - 1-8627 0-0629 11-65 40

1-3787 31 11 0-3726 2*0836 0-0416 - 1*7303 0-0706 13-15 40

1*4899 0 0-3739 2-3739 0 0 0-0745 13-93 40

1*3105 63 13 0-5576 1-4588 0-1123 - 1-4968 0Q1020 8-28 60

1-4599 52 37 0-5603 1-7746 0-0912 - 1-8385 0-1102 9 02 60

1*5894 42 3 0-5624 2-0475 0 0715 - 1-9179 0-1163 9-69 60

1-7721 21 0 05-648 2-4319 0 0348 - 1*3035 0-1235 10-63 60

1*8357 0 0-5655 2-5655 0 0 0-1257 10-95 60

2-0984 47 37 0-8522 2-2004 0-0926 - 1-8991 0-2000 8-01 90

2-1854 42 18 0-8533 2-3326 0-0814 - 19097 0-2038 8-31 90

2-3333 31 41 0-8551 2*5570 00599 - 1 7279 0-2090 8 83 90

2.4810 15 50 0-8566 2-7807 0-0294 - 10206 0-2136 9-27 90

2*5315 0 0-8571 283571 0 0 0 2151 9*45 90

of pressure seems to move down towards A' as a decreases, without any upward motion to A at all.

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Page 16: Resistance to a Barrier in the Shape of an Arc of a Circle

Resistance to Circular Barrrier. 431

v7. | a. PX. Pu | P. tan-P . AD/DA' MD/AA'.

(0-1493) - 00218 12 55 0 0280 0-0995 0 1034 74 18 2 47 0-2856 0-2778

(0-2477) - 0-0437 24 47 0-0883 0-1532 0(1768 60 3 5 10 0-3220 0-2564

(0- 3123) - 00658 35 19 0-1523 0-1661 0(2253 47 29 7 12 0-3592 0-2357

(0-3767) - 00658 50 19 0-2633 0*1860 0(3224 35 15 4 26 0-5511 0-1447

(0-0641) 0-0218 5 0 0 0085 0 1142 0*1145 85 41 0 41 0 7600 0.0682

(0 1200) 0-0437 10 0 0-0320 0-2031 0*2056 81 3 1 3 0*8119 0.0519

(0-2628) 0 0437 27 0 0*1377 0-2869 0 3182 64 21 1 21 0*7642 0-0668

(0*2820) 0-0437 30 0 0*1599 0-2924 0*3333 61 20 1 20 0-7668 0-0660

(0 .3756) 0-0437 50 0 0-3055 0*2653 0*4046 40 58 0 58 0 8253 0 0479

(0*4246) 0.0437 70 0 0 4082 0*1527 0*4358 20 30 0 30 0 9057 0-0247

(0 .4399) 0-0437 90 0 0*4522 0 0(4522 0 0 1.0000 0

(0. 1689) 0-0658 15 0 0-0667 0-2702 0*2783 76 8 1 8 0-8626 0-0369

(0-2117) 0-0882 20 0 0-1101 0-3197 0*3381 71 0 1 0 0-9085 0-0240

(0-2820) 0-0882 30 0 0*1789 0-3253 0*3713 61 12 1 12 0-8912 0*0288

(0 .3355) 0 0882 40 0 0 2523 0 3132 0*4022 51 9 1 9 0*8955 0-0276

(0 .4048) 0 0882 60 0 0 3791 0 2256 0(4411 30 46 0 46 0*9291 0 0184

(0 .4399) 0 0882 90 0 0 4645 0 0.4645 0 0 1 0000 0

(0-2820) 0 1340 30 0 0 2019 0 3521 0.4059 60 11 0 11 0-9890 0-0028

(0 .3355) 0-1340 40 0 0-2742 0-3310 0*4298 50 22 0 22 0 9781 0 0055

(0-3756) 0*1340 50 0 0-3398 0-2892 0*4462 40 24 0 24 0-9761 0-0060

(0-4246) 0 1340 70 0 0-4370 0-1613 0.4658 20 15 0 15 0-9850 0-0038

(0.4399) 0-1340 90 0 0*4719 0 034719 0 0 1*0000 0

(0.3571) 0-2071 45 0 0Q3357 0 3191 034632 43 33 - 1 27 1.0519 - 0 0126

(0.3756) 0 2071 50 0 0 3636 0 2920 0-4663 38 46 - 1 17 1 0458 - 0 0112

(0 .4048) 0-2071 60 0 0 4113 0 2292 0*4708 29 8 - 0 52 1 0307 - 0 0076

(0.4314) 0 2071 75 0 0-4632 0 1207 0 4786 14 36 - 0 24 1*0141 - 0-0035

(0.4399) 0-2071 90 0 0-4797 0 0-4797 0 0 1-0000 0

The - * - - curve represents the positions of the centres of pressure for the 1 imiting positions of concave barriers. It meets a 0 at MD/AA' = 0. 085.

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Page 17: Resistance to a Barrier in the Shape of an Arc of a Circle

432 L. Rosenhead.

Consider the graph of P against oc. It is at once seen that the effect of camber is much more marked when the angle of attack is small than when the angle of attack is large. This has been known since Lilienthal, who found that the thrust could be doubled and even trebled, at small angles of attack, by using a slightly cambered surface instead of a plane surface.

The following empirical formula for small cambers demonstrates this clearly: clerl

- sin M Ic + 20 2 + cos oc + 3cos2 oc 4+7rsin ta 9 sino (4+ -sin oc)T

General Summary. We have found the solution of the general problem of discontinuous fluid

motion past circular barriers. The results in the symmetrical case agree exactly with those found by Brodetsky. The plane barrier is dealt with as a particular case of the preceding theory, and the results obtained in this manner agree very closely with those obtained from the well-known exact solution. The approximations obtained by using the first three constants in the Levi- Civita expansion are found to be very good. Barriers convex to the stream are fully discussed, while barriers concave to the stream yield interesting results which are verified by experiment, especially the effect of camber on the thrust and on the motion of the centre of pressure.

In conclusion, I would like to place on record my sincere thanks and great indebtedness to Prof. Brodetsky, who not only suggested this problem to me, but also gave me the benefit of his advice and criticism throughout the whole of the investigations.

NOTE. The results obtained by Mr. Rosenhead are interesting from several points of

view. In the first place they demonstrate the power of Levi-Civita's method in the problem of discontinuous fluid motion. The accuracy obtained with the use of only a few coefficients A in the solution (1) above is quite remarkable. Prof. Levi-Civita remarked upon this circumstance when commenting on similar cases of discontinuous fluid motion at the International Congress for Applied Mechanics at Zurich, in September, 1926. We can consider any given barrier as defined by means of the radius of curvature R in terms of the direction of the tangent 0. We see that it is a comparatively easy matter to obtain coefficients Al, A2, giving IR in (5) such a relationship to 0 in (3) as is very close indeed to the given relationship. Thus some cases of the elliptic strut placed in a stream have been dealt with in this manner with considerable success.

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Page 18: Resistance to a Barrier in the Shape of an Arc of a Circle

Reststance to Circular Barrier. 433

In the case of the concave barriers discussed by Mr. Rosenhead, the remark- able effects of the camber deserve attention. It was to be expected that circular barriers concave to the stream would give increased thrust; but the increases obtained for small camber are strikingly great, especially with small angles of attack. A camber of 0 088 times the chord, or just over one-twelfth of the chord, is seen to produce an increase of over 60 per cent. with angle of attack 200. Even a camber of one-twenty-third of the chord gives an increase of about 70 per cent. at angle of attack 10'. But perhaps the most interesting point about Mr. Rosenhead's results is

with regard to the centre of pressure. The shift of the centre of pressure of an aerofoil is one of its important characteristics. The way in which the centre of pressure travels is seen to be dependent upon the camber. For cambers above a certain value, round about one-sixth of the chord, the centre of pressure moves backwards as the angle of attack is diminished. For smaller cambers the centre of pressure moves forwards as the angle of attack is diminished from 90?, reaches a stationary position, and then moves backwards as the angle of attack is further diminished. This becomes more marked the smaller the camber, as is evident from the continuous curves in fig. 4. The curve, consisting of dashes and dots, makes it clear that as the camber decreases to zero, the barrier becoming flat, the centre of pressure goes through a more and more marked forward and backward excursion. For zero camber, or 0-

the centre of pressure goes forward until the angle of attack is zero, reaching the position MD/AA' =0 188. But at the zero angle of attack the centre of pressure seems to jump back to MD/AA' about 0 085, if we consider the flat plate to be the limiting case of a circularly cambered barrier.

It would be wrong to make extravagant claims for the method of discontinuous fluid motion, but Mr. Rosenhead's work, as well as the results obtained by the present writer, show that it is perhaps more useful than is often supposed.

S. B.

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