The Flow Past a Rapidly Rotating Circular Cylinder

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The Flow Past a Rapidly Rotating Circular CylinderAuthor(s): M. B. GlauertSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 242, No. 1228 (Oct. 8, 1957), pp. 108-115Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/100386 .

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The flow past a rapidly rotating circular cylinderBY M. B. GILAUERT

Department of Mathematics, University of Manchester(Communicated by M. J. Lighthill, F.R.S.-Received 15 May 1957)

This paper considers,the two-dimensional flow past a circular cylinder immersed in a uniformstream, when the cylinder rotates about its axis so fast that separation is suppressed. Thosolution of the flow in the boundary layer on the cylinder is obtained in the form of a powerseries in tho ratio of the stream velocity to the cylinder's peripheral velocity, and expressionsare deduced for the value of the circulation and the torque on the cylinder. The terms calcu-lated expl]icitly are sufficient to give reliable numerical values over the whole range of rota-tional speeds for which the postulate of non-separating flow is justifiable.The previously accepted theory, due to Prandtl, predicted that the circulation shouldnot exceed a certain limit, while the present theory indicates that the circulation increasesindefinitely with increase of rotational speed. Strong arguments against the older theoryare put forward, but the experimental evidence available is inconclusive.

1. INTRODUCTIONThe flow with circulation past a circular cylinder is a classic example used in fluidmechanics for the discussion of the phenomenon of lift. The two-dimensionalirrotational flow, with an arbitrary value of the circulation, is elementary. For areal fluid, a circulation can be generated by rotating the cylinder about its axis,as is shown in the well-known series of photographs by Prandtl (Prandtl & TietjensI93 I). At small rotational speeds the photographs show a wide region of separationbehind the cylinder, but when the peripheral velocity of the cylinder exceeds twicethe stream velocity the separation largely disappears. Apart from the boundarylayer on the cylinder, the actual flow must then closely follow the irrotational flowsolution, with a certain value of the circulation.

Theoretical arguments have been given by Prandtl (1925; see also Goldstein1938, p. 82). If the stream velocity is U0,the velocity at the cylinder surface has themaximum value 2UO, or irrotational flow without circulation. If in addition theflow at the surface has a circulatory componelnt Q (so that the circulation K = rd9,where d is the cylinder diameter), then as Q increases, the stagnation points on thecylinder approach one aniother, coalescing when Q = 2UO. In this situation themaximum velocity at the surface is 4U0. If Qincreases further, the stagnation pointmoves off into the body of the fluid and closed streamlines encircle the cylinder.Prandtl argues that when the cylinder's peripheral velocity q is equal to 4U0, thecylinder nowhere exerts a retarding action on the fluid if Q = 2UO,but if Q wereless than 2Uo urther vorticity would be shed and so the circulation would increase;on the other hand, there is no tendency for Q to increase beyond the value 2Uoeven when q exceeds 4UO, ince convection of vorticity into the external stream isno longer possible. Prandtl thus predicts a maximum value of the circulation equalto 27rdUo,o be reached when q is nearly equal to 4UO. Since the lift L = pUOK,where p is the fluid density, this implies a maximum lift coefficient CL = L/( 2p U- d)equal to 41r.

The argument leading to the conclusion that Q never exceeds 2UO s open to[ 108 ]


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Theflow past a rapidly rotatingctrcular cylinder 109serious objection. Suppose that a regime is established with Q> 2U, so that thecylinder is surrounded by a set of closed streamlines, bounded by an outer dividingstreamline. If q is now increased above its previous value, vorticity of sign oppositeto that of the cylinder's rotation will be produced in the boundary layer. It is truethat this vorticity cannot be immediately convected downstream in the wake.However, if the increase in the value of q is sufficiently great, a body of fluid of largevelocity and vorticity will break away from the surface under the action of centri-fugal force and, after crossing the dividing streamline, will be carried away bythe stream. With a smaller increase in q, the vorticity will diffuse out through theregion of closed streamlines until it too crosses the dividing streamline and is con-vected downstream, for there is no mechanism to retain it close to the surface. Ineither case the effect is to lea-vethe cylinder with an increased circulation. In factit is clear that, for large values of q, the circulationi will increase until Q is approxi-mately equal to q.

It may be queried why lift coefficients in excess of 4r have not been observed inexperiments (Betz 1925; Prandtl 925; Reid I924; Thom I93 I). Almost certainlytheexperimental difficulties have been too great. Three-dimensional effects arise, evenon a cylinder of large aspect ratio fitted with end disks, since the vortex lines mustleave the cylinder at its ends and pass downstream. Wall interference effects are alsolikely to be serious, since the velocity due to the circulation falls off only as the reci-procal ofthe distance. In addition, it is possible that true two-dimensional flow wouldsuffer from instability in the neighbourhood of the stagnation point in the fluid.In this paper a theory is presented based on a study of the flow in the boundarylayer on the cylinder. The only assumption made is that the flow outside theboundary layer is the steady irrotational flow past a circular cylinder. A solutionof the boundary layer equations is obtained in the form of a power series in U0/q,andexpressions are deduced for the torque on the cylinder and the ratio Q/q. Sufficientterms of the series are calculated explicitly to give reliable numerical values overthe whole range of rotational speeds for which the assumption of non-separatingflow is justifiable.

2. THE BOIUNrDARY LAYERConsider a circular cylinder of diameter d, rotating about its axis with peripheral

velocity q in a stream U0.To study the boundary layer, choose fixed axes with xmeasured round the cylinder circumference and y normal to it. For conveniencein what follows, take x 0 to be the point at which the surface moves in the samedirection as the stream U0.On the assumption that outside the boundary layer theflow is everywhere irrotational, the velocity at the edge of the boundary layer is

U = Q+ 2Uocos (2x/d), (1)where Q is the circulatory component of fluid velocity. The boundary layer equa-tions for steady two-dimensional flow are

aana U dU a2,u (2)

_+_ o= - (3)0,


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110 M. B. Glauertwhere Iv s the kinematic viscosity of the fluid. The boundary conditions are

=U q, v=O at y=O, }(4)t6 U-> as y-*oo.

In view of (3), there is a stream function #f such thatu =4flay, v - avflx. (5)

It is convenient to introduce a complex number notation, rewriting equation (1)as U=Q{? +ace0}, (6)where 0 = 2x/d is the angular co-ordinate round the cylinder and

- x-2UO/Q. (7)Only the real part of equation (6) has physical significance. We are concerned withthe case in which Q is appreciably greater than U0,and hence ac s small. We mayobtain a solution in series of equation (2) by writing

f = y + cxf2(y) ei? + (X2{f(y) e210+ a2(Y)}

+ ax{f3(y)e310+ a3(Y) ei?}+ ac4{f4(y)40 +?4(y) e2'0+ h4(Y)} (8)Hence, from (5),

Q = 1+ acfIe1a+ C2ff2f210 g2}) ...U (9)2Q =- iaf e10 2ia2f2 e2i9 -.

In this expansion the circulatory fluid velocity Qhas been taken as the fundamentalparameter rather than the cylinder velocity q. For a study of the boundary layerequations this is the natural procedure, as the velocity at the edge of the boundarylayer is then completely specified. Fromi(9), u/Q -> I as ae ?0 for all y, which impliesthat q/Q-> 1 as Q-oo, as predicted in ? 1.

The major terms in equation (2) are those linear in ac.They are satisfied ifflI = y2(f _- 1), (10)

where y 2=Qi (11)vd'and hence Y = (1 +i),8, where V,- (Q/vd). (12)Equation (4) shows that the boundary conditions are

f1(0) = f (0) = 0, f )() , (13)and hence the required solutioni of (10) is

fj= 1-erU, fe1=y + (e-YY-1). (14)


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The flow past a rapidly rotating ctrcular cylinder 111Substituting in (9) and taking the real part, we thus obtain

u = Q+ 2Uo{cos0-e-fly cos (O-)}+ .. (15)The last term of (15) is precisely the velocity distribution in a shear layer, as occurswhen a plate oscillates harmonically in its own plane in a fluid at rest. This is asmight have been anticipated, since fluid elements, which are being carried roundthe cylinder with velocity Q, are experiencing a superposed velocity fluctuatingharmonically between - 2Uoand 2Uo.Care must be taken when considering the terms in equation (2) quadratic in a.If A and B are complex numbers,

98(A) (B)-= {1A(B +B)}, (16)where the bar denotes a complex conjugate. Consequently, terms independent of0 arise as well as terms proportional to e210.The resulting equations are

-l 'f2--f1 fl-~ (17)2 - 2 2 = 2fl -WiW-2 = 21fIfI+2flfI (18)y2The boundary conditions are

f2(O) =Af(O) = 0, f2(oo0) 0 , (19)92(?) =0, g2(0) =0. (20)No condition can be applied to y'(0), as it is known only that the surface velocityis independent of 0. Indeed, the value of g'(0) is one of the main objects of

our investigation, as it gives the first approximation to the amount by which qexceeds Q. Equation (18) may be integrated directly to give

2,2 =-2-flfl-v+ 27; (21)and further integrations are straightforward, on substituting from equation (14).Taking the real part of the solution, we obtain

g'2(O) = 2-fl, g2(O) = 1, g2(0o) = 3 (22)The integration of (17) is also straightforward, the required solution being

1 7Y1~~V2Y V2Cv( >12 12) (23)The next- term in the series (8) which gives a contribution to the difference

between q and Q is h4(y).This depends upon 3(y) but not on f3(y). The equationfor g3(y) iS72 g3-g3 = 12fl f2 + Wfl2-fl f2 + 2fl(q2 +g9) - f1(2flq+(g)2 (24)

with boundary conditions3(? ) = 0, 3g(?) = 0. (25)


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112 M. B. GlauertThe integration, although tedious, presents no difficulties. The equation for h4(y)may be integrated once to give

1 ~ 7 1 49I 3 1-h =,13- 913+fJ2J 49- 13 1 (26)y 4 fL 3--2fff 1 8-y 15 6 2 +15 2V )' (26,with boundary conditions h4(0) 0, hQ(o)= 0. (27)The constant in equation (26) is appropriate in view of the value of g3(oo), deter-mined in the solution of equation (24). Taking the real part, we have

h"(0) ( (28)and on integrating (26) we finally obtain

h4(0)-- 12 9 7 (29)

3. DisoUSSION OF RESULTSThe peripheral speed q of the cylinder is given by puttinig y = 0 in equation (9).

Using the calculated terms, we obtainq = I + a2g' (0) + (X4h4'(0) +..Q - 21

= 1+3( )-5.76() +.... (30)Whereas in the treatment of the boundary layer equations it was convenient totake the circulatory fluid velocity Q as the basic parameter, it is more naturalphysically to consider q to be the known quantity, and hence to invert equation(30), obtaining ( ) 4( ) (31)

From the computational point of view, equation (31) has two incidental advantagesover equation (30); the coefficient of the last term is reduced, and the denominatorsare larger, since q > Q. The variation of Q/Uowith q/U0, as given by equation (31),is shown in figure 1. Since the lift coefficient CL= (pU0K)/(1pU2d) and the cir-culation K = grdQ,it follows that Q/Uo = CL/27r, and so the curve of figure 1 alsogives the variation of lift coefficient with rotational speed. The last term of equation(31) is only 4 / of the first when q/U0= 3. For smaller values of q/U0 he assumptionof non-separating flow is suspect, so the terms calculated should be sufficient togive good accuracy over the whole range of rotational speeds for which the theoryis applicable.

In figure 1 are also shown the maximum lift coefficient as given by the theory ofPrandtl discussed in ? 1, and the curve obtained experimentally by Reid (l9z4).The experiments do not significantly confirm or contradict either theory, but theagreement is probably as good as could be expected in view of the technical diffi-culties referred to in ? 1. The fact that Reid observed a large drag force at the higher


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Theflow past a rapidly rotatingcircular cylinder 113rotational speeds is a clear indication that three-dimensional effects were sub-stantial.

Recently Wood (1957), applying an integral relation which he derives for a classof flows with closed streamlines, has obtained a formula for the circulation whichis equivalent to the first two terms of equation (31).

6 / /

>12? /~~~~~~~~4/

q/UO~ 4 1

FIGURE 1. Lift and circulation on a rotating cylinder (UO= stream velocity, q cylinder'speripheral velocity, Q = circulatory velocity at the edge of the boundary layer). (1) Pre-sent theory. (2) Prandtl's theory. (3) Reid's experiments. (4) Q = q (asymptote of (1)).The torque T on the cylinder is given by

T l td2f ( Mt),= (32)the negative sign being appropriate if T is considered to be positive when in thesense tending to increase the rotation of the fluid. Only terms of equation (8)independent of 0 contribute to (32), and hence we obtain, using the calculated valuesof ?2, T =lffd2Q{- 2g'(0) a4h4'(0)

= nTd2pUMR-W(Q02-2022(? + )(33)where the Reynolds number R = Uod/v. It is physically and computationallydesirable, as before, to express Q in terms of q by use of equation (31), so obtaining

T =7d2pU2R-i ((q) -0522( )+.J * (34)When qlUo= 3, the second term of (34) is 60 of the first, so once again the formula

8 Vol 242. A.


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114 M. B. Glauertshould be adequate over the whole valid range of rotational speeds. The measure-ments of torque by Reid (I924) were scanty, and were not extended to peripheralspeeds greater than UO;or q = UO,he value obtained was approximately

T = 5iTd2pU2R12.This is far above the value given by equation (34), but it is doubtful if it has nluchaccuracy or relevance, in view of the experimental imperfections and also theseparation occurring at this low rotational speed.

For energy conservation a torque on the cylinder is essential, to do work toreplace the energy lost by viscous dissipation in the boundary layer. However,from the point of view of angular momentum, it may excite curiosity that a torqueshould be necessary to maintain the steady irrotational flow past the circularcylinder. The explanation may be arrived at by a consideration of the displacementthickness S of the boundary layer. This is defined by the equation

#y - d) U (35)at the edge of the boundary layer. If

a = oc41(?) az242(o) * * - (36)the terms linear in oc in equation (35) show that

1 = -ei0. (37)

Taking the real part, 81= - cos(0- fir). (38)The implication of this is that, to the first order of approximation, the irrotationalflow is that past a circular cylinder of diameter d, displaced through a distancez/12f8 in the direction 0 = 17T.Now the forces exerted by the actual cylinder onthe fluid are the reversed lift force L through the centre and the torque T. Thesecombine to give a single force of magnitude L, with its line of action displaced fromthe centre of the cylinder by a distance e = T/L in the downstream direction.Approximating to T by the first term of (33), it is readily shown that e- =/2,8 + ....and thus the combined force acts through the centre of the displaced cylinder. Thepuzzle is thus resolved. The second term of (33) can also be explained by con-sidering the modifications to the shape and position of the cylinder represented byfurther terms in the expansion for the displacement thickness.

It might perhaps have been anticipated that 8, would be greatest at 0 = 0, wherethe velocity outside the boundary layer has its greatest excess over the surfacevelocity. However, it is a familiar result that in a shear layer the skin-friction isadvanced in phase by I7 over the velocity (of which it is the derivative with respectto y), and similarly the velocity has a phase-advance of 41T over the volume flux, towhich the displacement thickness is related. It is this phase-retarded componentwhich gives the effective cylinder a displacement in the stream direction, and impliesthe existence of a torque.


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Theflowpast a rapidly rotating ircularcylinder 115Finally, we may note that when employing the series expansion (8), there is no

need at any stage to make allowance for the effects on the pressure distribution ofthe curvature of the cylinder surface or the displacement thickness of the boundarylayer. These effects are proportional to an inverse power of the Reynolds number,while a, the ratio of successive terms in the series expansion, is independent of theReynolds number.

REFERENCESBetz, A. 1925 Z. Ver. dtsch. Ing. 69, 11.Goldstein, S. I938 Modern developments in fluid dynamics. Oxford University Press.Prandtl, L. 1925 Naturwissenschaften, 13, 93.Prandtl, L. & Tietjens, 0. 193I Hydro-und Aeromechanik. Berlin: Julius Springer.Reid, E. G. 1924 Tech. Notes Nat. Adv. Comm. Aero., Wash., no. 209.Thom, A. 1931 Rep. & Mem., Aero. Res. Coun., Lond., no. 1410.Wood, W. W. 1957 J. Fluid Mech. 2, 77.

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The Flow Past a Rapidly Rotating Circular Cylinder

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