Abstract
The article is an expanded version of the author’s report delivered at a meeting of the Göttingen Physical Society on November 17, 1924.
Full Text
THE MAGNUS EFFECT AND THE WIND-POWERED SHIP1
L. Prandtl.
I.
Flettner’s wind-powered ship is on everyone’s lips, thanks to an unusually zealous newspaper campaign. In connection with this, hydrodynamics—which until now had led a modest existence within a circle of narrow specialists—has suddenly acquired general interest. The “Flettner rotor” possesses such astonishing properties, which cannot in any way be explained by simple notions of wind pressure; suffice it to say that the force actions on a rotating cylinder must be 10–15 times greater than on a sail with the same visible surface.
Modern hydrodynamics can not only fully explain this phenomenon, mysterious at first glance, but it served as a systematic guide in discovering the large force actions of a rotating cylinder. The laws of hydrodynamics are insufficiently known in broad circles; they are, however, to a considerable degree a prerequisite for the correct explanation of the process. I therefore gladly take this opportunity, using the present phenomenon as an example, to introduce the reader as far as possible to the doctrine of the motion of liquids and gases.
But first of all we shall dwell a little on the history of the discovery of the remarkable phenomenon indicated. The effect was first observed not on a cylinder, but on a rotating sphere thrown into the air. Deviations of the line of flight of such a sphere from the ordinary trajectory of a thrown body were noticed long ago by artillerymen and by amateurs of ball games.
Even in those times when real round “cannonballs” were fired, artillerymen noticed irregular deviations from the ordinary trajectory of the projectile. Robins (B. Robins) as early as 1742 suggested that such deviations were connected with the rotation of the ball.
Subsequently he also gave an experimental proof of this. To regulate such entirely irregular rotations, around 1830 projectiles with an eccentrically located center of gravity began to be used. When firing such eccentric projectiles it was found that an “undershoot” occurred in the case when the projectile was placed in the gun with its center of gravity downward; the pressure of the powder gas was applied to the center of gravity of the projectile, and therefore a rotation of the ball was produced from above forward and downward. If the projectile was placed with its center of gravity upward, then an “overshoot” was systematically obtained. Accordingly, if the center of gravity was located to the right, then during the flight of the shell deviations to the right were obtained; with a “left” position of the shell, the deviations were to the left. Experiments in firing through several targets placed one behind another showed that these lateral deviations cannot be explained by lateral blows upon leaving the barrel of the gun, since the trajectory is clearly curved to the side and, consequently, the shell is constantly deflected by air forces.
For a definite solution of this question, the well-known Berlin physicist Magnus, Helmholtz’s teacher, carried out several laboratory experiments in 1852¹). One of these experiments was as follows. A brass cylinder could rotate between two points; rapid rotation was imparted to the cylinder, as in a top, by a cord. The rotating cylinder was placed in a frame which, in turn, could easily turn. A strong stream of air was directed at this system by means of a small centrifugal pump. The cylinder was deflected in a direction perpendicular to the air stream and to the axis of the cylinder, and moreover toward that side on which the directions of rotation and of the stream were the same. The direction of the deflection was the same as in the artillery experiments; the magnitude of the deflecting force Magnus did not measure; he supposed, however, that the order of this magnitude was the same as in the deflection of spherical shells²). Since then it has been customary to call the whole group of phenomena the “Magnus effect.” Thus the merit of Magnus, who first clarified the phenomenon in a laboratory setting, is duly appreciated.
However, even earlier than the artillerymen, players of ball games noticed the deflection of rotating balls. In tennis this phenomenon is very noticeable, and every experienced player knows that a “cut ball” differs markedly in its trajectory from one not cut³). If
¹) Cf. P. Pringsheim.—Naturwissenschaften, 13, 49, 1925.
²) In the same work Magnus describes experiments with rotating elongated projectiles and gives theoretical considerations on this subject. In popular notes, which can now often be found even in newspapers, these experiments, having nothing in common with the “Magnus effect,” are sometimes mixed together with it.
³) According to Vocker, already in 1671 Newton mentions that a cut tennis ball describes a spatial curve.
THE MAGNUS EFFECT AND A SAILING SHIP
to strike the ball obliquely from the right, then it deviates to the left from the plane determining its initial direction, and conversely. Balls struck from below fly farther; those struck from above fly nearer than they would along their normal trajectory. Still more remarkable, according to Walker, are the phenomena observed in the game of golf. In this case a ball struck from below flies for a considerably longer time and over a greater distance than would follow from the laws of falling on the basis of the initial velocity; moreover, the first part of the trajectory is noticeably concave upward. With an incorrect (side) blow, deviations to the side from the vertical plane determined by the initial velocity may reach 70 m.
The Magnus effect can easily be demonstrated without special apparatus in an auditorium or room. For this one may, for example, use a long cylinder glued together from paper. It is thrown while at the same time imparting to it a strong rotation (it is best to throw it forward, holding the hand in the same way as in bowling; the trajectory should be like that shown in Fig. 2). The phenomenon is especially clear if the cylinder is very long; in this case it is recommended to glue cardboard disks to the ends of the cylinder, which, on the one hand, increase its rigidity and facilitate rotation, and on the other hand constitute an aerodynamic improvement. In the middle of the cylinder a small paper projection is left, around which a cord with two ends is looped, as shown in Fig. 1. The ends of the cord are fastened to a wooden stick. If the cord is wound up and the cylinder released, it deviates to the side, as shown in the drawing.
Fig. 1. Experimental demonstration of the Magnus phenomenon.
Fig. 2. Experimental demonstration of the Magnus phenomenon (second experiment).
A paper triangular prism (in the present case with cardboard disks at the ends) likewise reveals such rotation very clearly if it is released while being held by the fingertips of one hand and given rotation by a finger of the other hand at the place indicated in Fig. 2 by an arrow1.
II.
Let us now turn to the experiments that explain the phenomenon. In artillery circles in earlier times there was a tendency to explain the phenomenon by saying that a rotating sphere experiences greater friction on the side of the “compressed air cushion” situated in front of the sphere, as a result of which a deflection to the side is produced. Poisson, however, showed that air friction is wholly insufficient to explain the phenomenon. The experiments mentioned above with eccentric spheres show that in reality the deflection occurs precisely in the opposite direction from what would follow according to the “cushion theory.”
In connection with his experiments Magnus proposed an explanation which, although it makes the phenomenon somewhat intelligible, can no longer be considered satisfactory; at that time hydrodynamics was still in a primitive state1. Considerably more is provided by the interpretation published in 1877 by Lord Rayleigh in connection with the question of the trajectories of sliced tennis balls. By that time hydrodynamics had undergone significant development in the works of Helmholtz, Sir W. Thomson, and others, though, to be sure, almost exclusively for the case of an “ideal fluid” without friction or compressibility. Lord Rayleigh’s calculations concern the flow of such an ideal fluid around a circular cylinder infinite in the axial direction. Rayleigh considers the state of the flow when a “circulatory flow” is superposed on an ordinary “potential flow”; he calculates the force exerted on the cylinder from the pressure distribution around the cylinder. Lord Rayleigh himself points out the weak point of his theory: the calculations are valid only for a fluid without friction. According to Thomson’s theorem, however, in the absence of friction a vortical motion cannot arise, or, if it existed, it cannot change. Thus the occurrence of the flow, which evidently must be connected with fric-
liquid, remained obscure. In any case, Rayleigh’s calculations are very instructive, and for us it will be useful to deal somewhat more fully with the theory of the motion of an ideal liquid; real liquids such as water and air possess only slight internal friction, so that in favorable cases conclusions about the character of the motion of an ideal liquid can give an approximate idea of the behavior of a real liquid. The question of whether we have before us a “favorable case” or not, of course, requires a special investigation. We shall return to this later.
Fig. 3. Potential flow around a straight cylinder.
The flows usually studied in an ideal liquid, caused by the motion of bodies in a liquid initially at rest or by pressures on surfaces, have the same geometrical character as flows of electricity in a homogeneous solid conductor, or as the magnetic field in a space of constant permeability. Speaking in the special language, these motions can be derived from a potential satisfying Laplace’s differential equation. An essential feature of such a potential flow is the absence of rotation of the separate elements of the liquid. This is closely connected with the absence of friction: without the aid of friction a particle of liquid cannot begin to rotate.
Fig. 4. Circulatory flow around a circular cylinder.
The two flows from which Rayleigh proceeds are shown in Figs. 3 and 4. A flow of the same type as in Fig. 3 can be obtained by passing an electric current through a tin plate with a circular aperture at the center. In doing this one must imagine that, along the edges of the poorly conducting tin, well-conducting copper strips are soldered, through which the current is supplied. The electrical analogue of the flow shown in Fig. 4 can be obtained only by cutting the tin plate with a circular aperture along one of the radii; the copper busbars supplying the current must be
at the same time attached along the edges of the slit that has formed on the right and on the left. The magnetic field of the type of Fig. 4, on the contrary, is well known. If an electric current goes perpendicularly to the plane of the drawing through the center of a circular opening, then the magnetic field will be exactly like that in Fig. 4.
Fig. 5. Superposition of two flows. From system of lines I and II there arises system III of diagonal curves.
The flow that Rayleigh considers is the result of the imposition, the superposition, of both flows (Figs. 3 and 4). In this flow the velocities at each point are obtained by addition, according to the parallelogram rule, of the two individual velocities. The potential is obtained by simply adding the two values of the potential at each point of space. The system of streamlines can be found if the two separate systems of lines are drawn one after the other in such a way that the quantity of fluid passing per unit time between two streamlines is everywhere one and the same; the resulting streamlines are obtained in the form of diagonal curves, Fig. 5. Judging by the intensity of the vortical motion, the superposition of flows (Figs. 3 and 4) leads to various forms; one such form for the case of moderate rotation is shown in Fig. 6, another, for stronger rotation, in Fig. 7.
For understanding what is achieved by such flows, we must deal with the distribution of pressure in the flowing ideal fluid. First of all, the pressure at one and the same point is identical in all directions, as in the case of a fluid at rest, but the pressure at two different points is, generally speaking, different. We shall not take into account the fluid’s own weight, understanding by the pressure in question only the difference
Figs. 6 and 7. Circulatory flows arising in the superposition of the flows of Figs. 3 and 4.
pressures in the case of motion of the liquid and at rest. This difference may be positive or negative; therefore we shall speak of increased and decreased pressure. If, on the path from some point \(A\) to point \(B\) along one and the same streamline, the pressure constantly decreases, then each particle of the liquid has behind it a greater pressure than ahead of it; therefore it will move, accelerating, in the direction of decreasing pressure. If at point \(A\) there was already a certain velocity in the direction toward \(B\), then this velocity will constantly increase on the way to \(B\); if, conversely, a particle of the liquid at \(B\) possesses a considerable velocity in the direction toward \(A\), then this velocity will slow down owing to the opposite pressure difference (at such a point the pressure ahead is all the time greater than behind). Consequently, in this case as well the velocity is greater at point \(B\), where the pressure is less, than at \(A\). Calculation for a stationary, i.e. time-independent, flow in an ideal liquid leads to the following relation: the sum of the pressure \(p\) and the quantity \(\rho \dfrac{v^2}{2}\) (\(\rho\)—density, mass of a unit volume, \(v\)—velocity) is constant along a streamline. This relation, established in 1738 by D. Bernoulli and often called “Bernoulli’s theorem,” is, of course, closely connected with the law of conservation of energy in the mechanics of a point. Let a ball roll in a smooth hollow (Fig. 8); at the deepest place of the hollow its velocity will be greatest, at the highest—smallest, while
Fig. 8. Motion of a ball under the action of gravity.
\[ h+\frac{v^2}{2g}=const. \]
(\(g\)—acceleration of gravity). As was to be expected, pressure in a flow plays the same role as the height \(h\) in the present case.
In stationary flows with a potential, moreover, the sum
\[ p+\rho\frac{v^2}{2} \]
is constant not only along a certain streamline, but throughout the whole region of the flow.
We shall now apply this to the flows in Figs. 3, 6, and 7. At point \(A\) of these flows the liquid is brought completely to rest for an instant; consequently, by Bernoulli’s theorem, the pressure here must be greatest—by the amount \(\rho \dfrac{V^2}{2}\) greater than in the undisturbed flow. (Here \(V\) is the velocity of the undisturbed flow relative to the body, or, in other words, the velocity of the body, or, in other words, the velocity of the body relative to the air at rest.) In the region of point \(B\), where the streamlines crowd together most of all and where, consequently, the velocity is greatest, the pressure must be lowest. At point \(C\) the pressure is as great as around \(A\). In the symmetric flow (Fig. 3) the pressure at \(B'\) is the same,
as also in \(B\); on the contrary, in the asymmetric flows, Figs. 6 and 7, the pressure in \(B\) is considerably less than in \(B'\), as a result of which there exists a force in the direction \(B'B\), which is precisely the Magnus force. From the distribution of pressures in all three flows one cannot detect any resistance in the direction of motion. This circumstance is connected with the replacement of a real fluid by an ideal one: to overcome resistance, work must be expended, the equivalent of which can only be the kinetic energy of the fluid. If, however, the flow is the same in front of the body and behind it, then there can be no such kinetic energy, and consequently no resistance, here. However, Relê’s calculations, and consequently also our last considerations, are valid only for a very long cylinder, in which one may disregard the conditions at the ends. For a short cylinder this is no longer permissible. Here one can only mention that even in an ideal fluid, in rotational motion, at the ends of the cylinder there appears a kinetic energy associated with the vortex; it remains in the flow, and here a corresponding resistance arises (the so-called induced resistance, as also in lifting wings). Hence it is permissible to conclude—and experiments confirm this—that the Magnus effect can be fully observed only with very long cylinders; with short cylinders and spheres the effect is observed in a form strongly distorted by vortex motion1. All the older observations were made with relatively short bodies; the measurements in Göttingen in 1923 were the first to be made on sufficiently long cylinders.
Two further remarks are appropriate here. First, the pressure distributions considered can also be readily represented in the following way. If a particle of fluid moves along a curved streamline, it is easy to understand that, for this, a force must act on the particle deflecting it toward the concave part of the path. It follows that the pressure on the convex side must be somewhat greater than on the concave side. One may also say that the particle, in its tendency to move rectilinearly, presses on the convex side, and consequently a “centrifugal force” appears. Tracing these pressure differences in the direction transverse to the streamlines as far as the region of undisturbed flow, where the pressure is equalized, we again find that in region \(A\), where the streamlines are most convex toward
cylinder, there is an excess of pressure, while in region \(B\), where the lines are most concave toward the cylinder, there is reduced pressure. If one carries out a quantitative calculation on the basis of this representation, the same pressures are obtained as from Bernoulli’s theorem, which is clear from the internal connection of this representation with Bernoulli’s theorem.
The second remark concerns the magnitude of the Magnus force. On the basis of Rayleigh’s pressure distribution one finds a formula according to which this force is proportional to the product of the velocity \(V\) of the cylinder relative to the undisturbed fluid and the velocity of the circular flow \(U\) (Fig. 4). For a segment of an imagined “infinitely long” cylinder of length \(l\) and radius \(r\), this force is
\[ P=\rho VU\cdot 2\pi r l, \]
where \(\rho\) is the density of the fluid.
The velocity of the circular flow \(U\) is often erroneously confused with the peripheral velocity of the rotating cylinder. The relation between these velocities is not determined in advance and is in general not simple; for the time being it has to be found from experiment. In particular, in Fig. 6 \(U=V\), and in Fig. 7 \(U=2V\).
It should be noted that Rayleigh’s formula is a special case of Zhukovsky’s formula (1906)
\[ P=\rho VTl, \]
which is applicable to all cases where the flow gives rise to a lateral force, i.e. to lifting wings, sails, etc. Here \(T\) is the “circulation,” obtained as follows: each linear element of any closed curve enclosing the body that gives rise to the force is multiplied by the component of velocity in its direction, and these products are summed (integrated). Such a “circulation” has remarkable properties in the case of potential flow. In ordinary flows, as, for example, in Fig. 3, it is equal to zero for any closed curve; for circular potential flows of the type shown in Figs. 4, 6, and 7, the “circulation” for any closed curve not enclosing the body is likewise equal to zero, but for any curve encircling the body once it has a finite magnitude, so that its value \(\Gamma\) is a measure of the circular flow. Let in Fig. 4 the radius of an arbitrary circular streamline be \(r\); the velocity \(u\) must be taken in full, since its direction coincides with the linear element; consequently, in the present case \(\Gamma=2\pi r u\); the magnitude \(\Gamma\) is constant, and therefore \(u\) must be inversely proportional to the distance \(r\).
The concept of circulation makes it possible to formulate more precisely the important theorem of W. Thomson mentioned only in passing above. This theorem states that in a homogeneous frictionless fluid
circulation, taken along any line permanently consisting of the same particles of fluid, cannot change with time. This theorem, under the indicated conditions, remains valid not only for “potential motions,” but also for any vortex motions. If at first the cylinder is moved without rotation, then no circulatory motion arises (Fig. 3), but if the cylinder is then set into rotation, it is entirely unclear how, suddenly, contrary to Thomson’s theorem, circulation can arise. Thus the state of affairs, despite the very satisfactory picture of the flow and of the pressure distribution, is quite hopeless from the point of view of an ideal fluid; it remains incomprehensible how a rotational motion of the fluid can arise.
If one looks more closely, our solution for the simple flow around a nonrotating cylinder (Fig. 3) is not very satisfactory either. It is known that in an actual fluid such a cylinder moves by no means without resistance, and the forms of the flow in a real fluid are completely different from those shown in Fig. 3; in reality, vortex motions arise behind the cylinder. We shall see later that the explanation of the deviation of a simple flow with resistance from the ideal picture (Fig. 3) also gives a key to explaining the Magnus effect.
III.
One can indicate the reason for the helplessness of the theory of an ideal fluid with respect to the problems mentioned. The forces of friction in such slightly viscous fluids as water and air are negligible within the fluid; they may be neglected in comparison with the forces of inertia; but in thin layers immediately at the surface of immersed bodies, or of solid bodies, these forces become of the same order of magnitude as the forces of inertia. If one imagines that the friction (viscosity) of the fluid continuously decreases, then the specific manifestations of friction at the surface do not diminish; only the thickness of the corresponding layer narrows.
It is not difficult to convince oneself of the presence of such a layer: all experimental measurements with viscous fluids show that the layer directly adjacent to the bodies adheres to them, as it were, i.e. is at rest relative to them. The following layers move one relative to another, so that the velocity of a layer farther from the wall is greater than that of a layer situated closer to it. Thus, around a body or wall there is formed a zone in which the velocity changes from zero immediately near the wall to the velocity of the free flow, on which friction has no influence. Such a transition of velocities is effected by the forces of friction; these forces, calculated
on a unit volume of the same order of magnitude as the pressure forces caused by the effects of the inertia of the free fluid, the velocities in the friction zone differ by finite amounts from the velocities of the free fluid. The character of the distribution of velocities in the friction zone is shown in Fig. 9. The “thickness” \(\delta\) of this zone may be taken as approximately equal to
\[ \frac{1}{50} - \frac{1}{100} \]
of the diameter of the cylinder, depending on the greater or lesser viscosity1.
Next, evidently, the question arises of the laws of motion in this friction zone, or “boundary layer,” as specialists say (the expression is not altogether apt, but it has been accepted, and therefore we shall use it). These laws are to a considerable extent amenable to calculation, but the calculations themselves are very difficult.
The most important results, however, can be understood from a qualitative consideration. On the particles in the “boundary layer,” on the one hand, as in the free fluid, there act accelerating or retarding pressure differences; on the other, they are slowed by friction at the wall. Let us look at a concrete example to see what results from such an interaction. For this purpose let us choose the beginning of the motion of a circular cylinder which until then had been at rest2. For the free fluid the laws of an ideal fluid are applicable with sufficient accuracy. According to our supposition, at the beginning everything was at rest; therefore at the beginning, for every closed line, the circulation is equal to zero and must thereafter remain equal to zero for all lines formed by the same particles of fluid. Therefore at first, regardless of whether the cylinder that has begun to move begins to rotate or not, only a potential flow of the type of Fig. 3, without circulation, is possible, possessing, as we saw above, this property. Suppose that the cylinder does not rotate, and let us examine the friction zone. If the acceleration under changing pressure conditions has been fully attained and the cylinder moves uniformly, then at \(A\) and \(C\) (Fig. 3), as was said above, there is increased pressure, and at \(B\) there is decreased pressure. On the path from \(A\) to \(B\) the particles of the free flow acquire kinetic energy and expend—
Fig. 9. Velocity distribution near a wall.
catch it again on its path from \(B\) to \(C\). But in the boundary layer, as a result of friction against the wall, the particles lose part of their kinetic energy and are no longer capable of scattering in such a way as to penetrate into the region of high pressure at \(C\); they stop and, owing to the pressure drop from \(C\) to \(B\), return back. The relations here are the same as for a ball in a wave-shaped channel (Fig. 8), if it were somewhat retarded by friction; not having reached \(C\), the ball will turn back. In the boundary layer the conditions are somewhat different from those for the rolling ball, because (we have not yet spoken of this) the free fluid exerts on the boundary layer a certain force that pushes it forward. Thanks to this force, among other things, the reverse motion is less than it would be in its absence1.
In particular, the following occurs: the most strongly retarded inner layers first move in the reverse direction, the neighboring layers follow them, and only the outermost layers of the friction zone are carried farther along by the outer stream. The boundary layer at \(B\) holds back ever new quantities of fluid, which also move back; thus, between \(B\) and \(C\) there is formed a compacted lump of fluid, which acquires rotation as a result of friction; this lump, under the influence of the pressure difference, moves toward \(B\) and, colliding with the stream advancing forward, enters the free fluid in the form of a “vortex.” Thus, gradually, owing to imperceptible processes in the boundary layer, the flow is completely altered. The flow separates completely from the wall near \(B\), forming ever new vortices; between it and the wall there remains a region with irregular weak motion.
Several photographs, obtained by me about twenty years ago, when I first took up these matters, with the aid of primitive apparatus, may illustrate the process more clearly. In a small channel water flows, set in motion by a wheel with blades; particles of specular iron are stirred into the water. This is a red mineral consisting of thin shiny flakes; in the vortical motion of the water the scales are oriented differently in different parts of the vortex and cause different reflection of sunlight. In Figs. 10, 11, and 12 three different states of the flow around a cylinder are shown; the first picture corresponds to a very short path, the second to a somewhat longer one, and the third to a still longer one. Fig. 13 shows the established state, which is characterized by an oscillatory motion of the vortex wake.
At the same time, twenty years ago, I was also able to give a visual proof that the cause of vortex formation is the processes
in the boundary layer. If, at the place where the reverse flow first forms, a slit is made in the cylinder through which water can be continuously sucked out, then in this way the retarded
Figs. 10–13. Flow around a circular cylinder in various phases of development.
fluid can be removed before it begins its reverse motion. The effect of such suction is seen in Figs. 14 and 15. In the figures one can notice the end of the rubber tube carrying away the water being sucked out; on the side of the cylinder where pumping is performed, no vortex is formed, and the flow
Figs. 14–15. Flow around a cylinder with suction.
does not separate. It is also noteworthy that the separation of the flow, now absent at the cylinder, occurs at the straight walls of the channel.
The condition for separation of the flow is connected in this case not with the convex shape of the wall, as at the cylinder, but with the fact that without this separation
there would be a slowing of the motion (accompanied by an increase in pressure). If circumstances are such that, at a straight wall, such an increase in pressure must arise, then a reverse flow appears, then a vortex, and then a deflection of the flow away from the wall. It should also be noted that the beginning of the formation of a vortex at the wall of the channel is already indicated in Fig. 14 at point a. If a slot were made here in the wall, one could perceive separation; in that case there would be, at this point, a constant increased pressure and a slowing of the flow1.
At that time I did not carry out experiments with a rotating cylinder. For external reasons (in the autumn of 1904 I moved from Hanover, where the described experiments were being carried out, to Göttingen; here at first I had other tasks; besides, it was necessary to build anew an apparatus similar to the Hanover one), only later did I return to experiments with water flows (beginning in 1907), and I investigated, among other things, the flow around two cylinders rotating in opposite directions and touching one another in the stream. In this case one might have expected that, with sufficient speed of the cylinders, the formation of vortices and separation of the flow would cease, since here, owing to friction against the walls moving together with the flow, the fluid is not slowed down, but in the extreme
Fig. 16. Two cylinders rotating in opposite directions.
1 Recently, in the experimental institute under my direction, experiments have again been begun with suction at walls. It was found that if many narrow slits are arranged along the wall, through which the fluid is sucked off in small quantities, then a number of flows is obtained that differ very greatly from the usual ones (for example, a deflection of an air flow by 180° under suction in a pipe). Clearly, the suction method is applicable in all cases where separation must be avoided. For example, for lifting wings, sails, turbines, propellers, ship hulls, turbine blades, pipes, diffusers, etc.
in this case, on the contrary, is accelerated. The experiment confirmed this. The walls and bottom of the channel were covered with moving sheets, so that here too the separation of the flow was avoided. These sheets, moved by means of shafts, nevertheless caused a strong distortion of the motion. A photograph of such a flow is given in Fig. 16¹). In connection with these experiments, in one case a single rotating cylinder was also studied, although at that time no special significance was attached to this experiment. In Fig. 17 the only surviving photograph is reproduced. These photographs were obtained by the method of Prof. Ahlborn in Hamburg, with illumination by an electric spark, and were technically very imperfect. Later the photographic technique was considerably improved. In Figs. 18–21 a series of photographs is reproduced,
Fig. 17. Rotating cylinder.
obtained by Rubach in 1913–14. They very clearly show the processes of separation near steep cylinders. At first there is potential flow; only in a very narrow zone are the beginnings of a reverse flow noticeable. But a pair of vortices grows rapidly; at the place where they touch the cylinders they cause secondary processes of separation and vortices. These secondary vortices increasingly complicate the picture, which becomes irregular and finally turns into an oscillating flow accompanied by the formation of new vortices. Photographs of flows with rotating cylinders were not made at that time. I hope to obtain such photographs, as well as photographs of the processes of suction, in the near future.
¹) The small vortices behind the pair of cylinders arise because the boundary layer here moves faster than the rest of the flow; they have nothing in common with the vortices that lead to separation of the flow. The cloud-like disturbances of the same origin on both sides of the flow sprinkled with lycopodium are caused by walls moving faster than the water.
The preceding exposition makes it possible simply to explain the emergence of the circulating flow, for rotating cylinders, that is necessary for a satisfactory theory of the Magnus phenomenon.
Fig. 18.
With sufficiently strong rotation, on the side of the cylinder that moves in the same direction as the flow no braking will occur, and vortices will not arise and break away; on the contrary, on the other side a vortex will arise, just as happens in the cylinder with suction. For the line \(abcda\) in Fig. 22, which encloses both the cylinder and the vortex in the region of free flow, the circulation is still equal to zero. If two lines \(ba\), traversed in opposite directions, are added, then nothing changes: these two portions of the contours mutually neutralize one another. But from the lines considered one can compose two further closed paths, \(abda\) and \(cdbc\). For the latter line, which encloses only the vortex, the circulation is different from zero; consequently, the line \(abda\), enclosing the cylinder, must also have the same circulation. The vortex drifts away with the flow, but the circulation of the cylinder remains\(^1\). With weak rotation of the cylinder only one of the two
Fig. 19.
\(^1\) It should be remembered that Thomson’s theorem ceases to be valid in the frictional zone; therefore closed lines that pass anywhere through the fluid originating from the frictional zone have a circulation different from zero.
of the vortices will be smaller than in the case of a stationary cylinder, while the other vortex will be larger. Here the circulation around the cylinder will be equal to the difference between the circulations of the departing vortices.
Let us note, as a warning against a widespread error, that the kinetic energy of the circulation flow has nothing in common with the work of air friction which the cylinder must overcome during rotation. My colleague, Eng. Ackeret, has shown (in a work not yet published) that the cylinder has to overcome resistance as it advances while the circulation is being formed; the corresponding work is the equivalent of the energy of the flow that arises.
Fig. 20
Ultimately, the friction of the air performs a regulating function; if, for example, the circulation, when the speed of rotation of the cylinder or the wind speed changes, ceases to correspond to the “normal” state, then, as a result of air friction, more vortices of one direction of rotation than of the other are formed, until the circulation corresponding to the given instantaneous state is established.
Fig. 21. Formation of vortices around a cylinder.
Photographs by Rubach.
From the point of view of this information about the boundary layer, we shall once again examine more attentively the flow in Fig. 7.
L. Prandtl
Here the flow goes around the cylinder only in one direction. If the cylinder begins to rotate with a peripheral velocity exceeding the maximum velocity of the flow, then the boundary layer will not be retarded anywhere; on the contrary, it will everywhere be carried forward, and therefore here one can no longer expect separation of the vortex when the circulation corresponding to this flow is reached. Hence we conclude: 1) that in this case the picture in Fig. 7 approximately corresponds to reality also in the case of the rotation of a cylinder in a viscous fluid; 2) that the corresponding transverse pressure difference corresponds to the theoretical maximum1. What must be the peripheral velocity of the cylinder when such a state is reached? First of all, the theory gives, for the maximum velocity of the flow in Fig. 3 (at \(B\) and \(B'\)), the value \(2V\); the velocity of circulation \(U\) of the additional flow in Fig. 4 is therefore also \(2V\), so that at \(B\) and \(B'\) there arise velocities \(4V\) and \(0\); the same reasoning is applicable also to peripheral velocities of the cylinder \(u\) greater than \(4V\). It may, however, be expected that it is permissible to deal also with velocities somewhat smaller than \(4V\); a slight retardation at the point of maximum flow velocity, obviously, can do no harm, since it is equalized by the pressure at places with a smaller flow velocity.
Fig. 22. Origin of circulation.
On the basis of our calculations in Section II, one can also determine the maximum theoretical force.
\(\Gamma=2\pi rU=4\pi rV\), consequently
\[ P_{\max}=4\pi\rho V^{2}rl. \]
In order to pass to the number used in aerodynamics, \(c_a\) [the lift coefficient (Auftriebzahl)], we divide \(P\) by the visible surface of the cylinder
\[ F=2rl \]
and by the “head height” \(q=\rho\dfrac{V^{2}}{2}\); we obtain
\[ (c_a)_{\max}=\frac{P}{Fq}=4\pi=12.57^{\,2}). \]
It may be noted that this drag is approximately 10 times greater than the values attained on ordinary aeroplane wings. This is connected with the fact that the flow around a rotating cylinder is deflected considerably more strongly than at a lifting surface. If one studies the distribution of the flow pressure in Fig. 7, then Bernoulli’s theorem leads to the conclusion that at points \(A\) and \(C\), in comparison with the undisturbed flow, there is an excess of pressure corresponding to the ordinary “velocity head” \(\rho \dfrac{V^{2}}{2}\), but in the region \(B\) the pressure is less by \(16 \rho \dfrac{V^{2}}{2}\) than at \(A\); therefore the pressure deficiency, in comparison with the pressure of the undisturbed flow, is equal to fifteen times the “velocity head.” Thus the main part of the transverse force is achieved by suction effects! This is also clear from consideration of Fig. 7, if one recalls the centrifugal effects in a liquid, which are, evidently, very considerable in the region situated above the cylinder1.
IV.
For a long time experiments with a rotating cylinder, spheres, and other bodies had been entered on the list of current tasks of the Aerodynamic Experimental Institute under my direction; the theoretical considerations indicated above especially prompted an experimental verification. Other urgent tasks, however, compelled us to postpone the investigation. In the spring of 1923 we obtained very high-speed small electric motors, developed by my long-time and extraordinarily valuable collaborator Betz for rotating the propellers of aeroplane models2; this served as the final occasion for beginning the work. H. Ackeret, who was very interested in all questions connected with the boundary layer, then undertook the investigation of rotating cylinders. In order to approximate the theoretical conditions as closely as possible, the cylinder was placed between two parallel walls; in this way the processes associated with the flow should be identical in all planes parallel to the two walls, and harmful flow around the ends of the cylinders could be avoided. The peripheral speed was brought up to four times the wind speed, but at first the highest value attained for \(c_a\) was about 4 instead of the theoretical value 12.57. Investigation of the flow soon showed that only the middle part of the cylinder was working properly; at the sides the flow did not remain attached to the cylinder and therefore was deflected little. I attributed such
disagreement with the expected flow separation of the air current on both walls, corresponding to separation on the side wall in Fig. 15. To avoid this I proposed placing, at the ends of the cylinder, disks rotating together with the cylinder; they were to prevent the braking of the boundary layer at the critical places (cf. Fig. 23). Expectations were justified. The flow spread from the cylinder to the walls, and the number \(c_a\) increased approximately to 10 when the air current, about 20 cm high, was deflected almost at a right angle by a cylinder 4 cm thick. This could be considered satisfactory, since one could not expect the value 12.57 to be attained because of deviations caused by friction.
Considerations on the practical use of the rotating cylinder had already been expressed by us in connection with the theoretical results, but none of the indicated possible applications (airplane wings, propeller \({}^{1}\), windmill wings, turbine blades, etc.) proved sufficiently advantageous. In this respect I have not changed my views even now. By using rotating cylinders instead of wing-like forms, one can considerably reduce the depth measured in the direction of the flow, since on the visible surface of the cylinder there acts a force 8–10 times greater; but this circumstance is not of great interest if one takes into account that the realization of forms without moving parts is much simpler; on the other hand, good forms of lifting wings experience less resistance in the direction of motion. Therefore it is important to note that, by using rotating cylinders, one cannot save on the area of airplane wings, windmills, etc., since the power limits of these parts depend to a considerable degree on the quantity of air captured per unit time. This quantity, however, is determined by the area of the wings.
Fig. 23. Diagram of the experimental setup with a rotating cylinder.
Unfortunately, we did not then examine the case of ship sails, where there are other ratios and where rotating cylinders are advantageous. This was done by A. Flettner, the well-known inventor of the “Flettner ship rudder.” In the institute under my direction he had already earlier carried out various experiments for
\({}^{1}\) Prof. Gümbel (Gümbel) in Berlin, as early as 1918, tested a propeller with rotating cylinders.
THE MAGNUS EFFECT AND THE WIND-DRIVEN SHIP
...to ascertain the properties of his invention and came to the idea of applying the idea of this rudder1 to a ship’s sail. For economic reasons, sailing vessels gradually gave way to steamships and ships with Diesel engines; the chief reasons for this were the need for a large crew to handle the sails, and frequent work on repairing the rigging. Flettner wanted to introduce metal sails, constructed like the lifting wings of metal aeroplanes; the sails were to be set to the wind automatically by means of auxiliary rudders. Storms presented a difficulty. Metal sails cannot be furled, but with the aid of auxiliary rudders they can be set exactly in the direction of the wind, without experiencing lateral pressure. But what is to be done if, during a storm, the auxiliary rudder suffers an accident and remains in a position corresponding to full pressure on the sail?! Further disappointment was brought by a comparison of the new sails with the old ones; it turned out that ordinary sails, when properly set to the wind, were by no means as bad as people had been inclined to think; the forces obtained reached 80 percent of the magnitude measured with metal sails. For it to be advantageous to replace the old sails by new ones, the metal sails would have had to be made extraordinarily large. Flettner therefore turned to the search for another solution. When he was told of the Göttingen experiments with a rotating cylinder, he quickly decided to investigate the applicability of such cylinders to his sailing ship, and for this purpose entered into contact with us. On the basis of our preliminary experiments we were then able to propose to him, as the most advantageous form, the one that was in fact realized on the ship. According to what has been set forth above, the cylinder must be long and have disks at the ends. The upper free disk must now serve another purpose than the disks at the walls of which we spoke above. Without this disk, air would penetrate from the end surface of the cylinder into the region of reduced pressure and would thus, over a considerable part of the length of the cylinder, destroy the circulation flow; such destruction will be the greater, the more considerable the reduction of pressure. Of course, the disk must rotate together with the cylinder in order to avoid the separation of the flow around it, of which we spoke earlier. These disks also have another advantage, clearly revealed in experiment: they reduce the induced-
resistance, owing to the fact that the edge vortex is divided into two vortices departing from the edges of the disk; the advantage here is the same as in the transition from a monoplane to a biplane1.
Fig. 24. Rotating cylinder with motor.
Now we shall briefly report on the further experiments. First, a cylinder with an electric motor placed inside it was investigated (Fig. 24), without disks and with two pairs of disks of different diameters. In Fig. 25 a curve is given showing the relation between the “lift coefficient” \(c_a\) and the “drag coefficient” \(c_w\)2 in the form of a “polar diagram”; in the lower left corner a dashed polar diagram of an airplane wing is plotted. On the curve in Fig. 26 the dependence of \(c_a\) on
\[ \frac{u}{V} \]
(the ratio of the peripheral speed of the cylinder to the wind speed) is shown. We see that in the region
\[ \frac{u}{V}=4 \]
the greatest lift is attained for a cylinder with disks, \(c_a\) approximately reaching the value 10; a cylinder without disks reaches only \(c_a=4\).
Fig. 25. Polar curve: \(c_a\)—lift coefficient, \(c_w\)—drag coefficient.
In addition, the wind force was investigated on a model of the ship “Bukau” with a rotating cylinder and simultaneously on a sailing ship of the former type. In diagram 27 these forces are shown for a relative wind3 of constant direction
and forces in such a way that the useful component in the direction of the ship’s motion was plotted for all possible course directions relative to the wind. The sail areas of both models (Fig. 28) were in the ratio \(1:9.8\). On the sailing model the sails, when the course was changed,
Fig. 26. Dependence of \(c_a\) on \(\dfrac{u}{V}\).
had to be reset each time; depending on the quality of the setting, the measured points obtained lay either inside or outside the diagram. The range of scatter of these points is indicated on the diagram by hatching. Adjustment in the “rotor” is possible only in the form of a change in speed, depending on the strength of the wind. The absence of any resetting when the wind direction changes is an important advantage of the “rotor.” On a sailing vessel, every change in the direction or strength of the wind is associated with a change in the position of the sails. On large ships this is especially tiresome, and therefore the sails are not turned until the vessel is somehow moving; as a result, very poor sailing is obtained. In rotor ships the correct setting is obtained by itself; to change the number of revolutions to the required value requires only minimal effort—for this the helmsman need only turn the handle of the electric motor rheostat. Only when the wind changes from the bow to the stern is it necessary to change the direction of rotation of the rotors.
Fig. 27. Wind forces on ship models, —— rotor; ······· sailing vessel.
If the cylinders are made to rotate in opposite directions, the ship can be turned on the spot. As the wind increases, the ratio \(\frac{u}{V}\), and with it \(c_a\), decreases, i.e., the force of the wind increases more slowly,
Fig. 28. Two ship models. Each rotor has an electric motor inside (construction as in Fig. 24).
than on old sailing ships, where the sails have to be taken in. The force can be weakened still further by reducing the number of revolutions. If, during a severe storm, the rotors are brought completely to a stop, then the pressure
Fig. 29. The “sails” of the “Buckau” before and after conversion.
THE MAGNUS EFFECT AND THE SAILING SHIP
of the wind remains very small, \(c_a=0\) and \(c_w=0.3\) ^1). The resistance in this case is less than that of the bare rigging of an equivalent old sailing vessel.
In Fig. 29 the motor sailing ship “Buckau” of 600 tons is shown schematically before and after reconstruction (the old sails are indicated by dashed lines). Fig. 30 gives the view from the captain’s bridge aft toward the forward rotor, the details of which are clearly visible in the photograph. The mast in the middle of the ship serves for lifting loads being brought aboard.
During the trial voyage of the “Buckau” on November 12, 1924, I was able to satisfy myself of the extraordinary care and beauty of the construction of the rotors, as well as of their machinery, at the Kiel shipyard “Germania.” Inside the rotors there is a column of sheet steel rigidly connected with the ship’s deck; it supports the axle on which the cylinder hangs. At the bottom the rotor has a second support. The motion is transmitted by a pair of gears placed just above the upper axle, which is set in motion by an electric motor according to Leonard ^2). The rotors are made of sheet iron 1 mm thick and, for greater rigidity, also have an internal structure. Practi-
Fig. 30. View from the captain’s bridge of the “Buckau” toward the forward rotor.
^1) The larger value of \(c_w\) in Fig. 25 is connected with the fact that, when the model cylinder was stopped, the critical speed (more precisely, the critical Reynolds number) had not yet been reached. Cf., for example, Wieselberger, Phys. ZS. 22, 321, 1921; L. Prandtl, Festschr. d. Kaiser Wilhelm Ges. Berlin. 1921, p. 178; Ergebn. d. aerodyn. Versuchsanst., II issue, Munich, 1923, p. 23.
^2) In Leonard’s scheme the direct-current motor is driven by a special dynamo connected with it in such a way that the magnetic field of the motor is excited by an external direct current; the field of the dynamo, however, can be regulated; both armatures
they operate practically noiselessly; in the opinion of specialists, the ship’s maneuvering capability proved excellent. The ship has not yet been tested during a storm, since there has been no storm since the completion of construction. In this respect there is no need for apprehension, since the forces of the wind with the rotors stopped are very small1.
Naturally, the main question arises: can the rotor ship compete economically with steamships and motor vessels? Calculations—which, however, lie outside the limits of my competence—apparently confirm this. Real proof, of course, can be provided only by experience with rotor ships. In practice much will become clear that cannot be taken into account in advance (repair costs, etc.!). Yet in this respect as well, one may suppose, the prospects are favorable; therefore it is very gratifying to hear that a number of large motor sailing vessels are to be rebuilt as rotor ships. Then the test can be carried out on examples. It is only a pity that once again the machine is driving out of life a little corner of poetry. However, even without this it would not have been possible to save sailing ships. Let us wish success to the successor of the sailing ships!
LITERATURE.
1) B. Robins. New principles of gunnery. London. 1842.
2) B. Robins. Mathematical tracts of gunnery. London. 1761, pp. 200 ff.
3) S. D. Poisson. Recherches sur le mouvement des projectiles. Paris. 1839.
4) I. P. V. Heim. Beiträge zur Ballistik in besonderer Beziehung auf die Umdrehung der Artilleriegeschosse. Ulm. 1848.
5) G. Magnus. Über die Abweichung der Geschosse. Abh. d. Kgl. Ak. d. Wiss. zu Berlin. 1852: Pogg. Ann. 88, 1, 1853.
6) J. W. Strutt (Lord Rayleigh). On the irregular flight of a tennis-ball. Messenger of Mathematics 7, 14, 1877, Scientific papers. Cambridge. 1899, p. 344.
7) G. t. Walker. Article “Game and Sport” in Enzyklopädie der math. Wissensch. IV, 9, p. 136 ff. 1900.
8) C. Cranz. Article “Ballistics” in Enzykl. der math. Wissensch. IV, 18, p. 226 ff. 1903.
9) L. Prandtl. Über Flüssingheitbewegung bei sehr kleiner Reibung. Verhandl. d. 3-ten Intern. Mathematikerkongr. zu Heidelberg, 1904. Leipzig. 1905, p. 484.
10) Lafay. Sur l’inversion du phénomène de Magnus C. R. 151, 867. 1910.
11) Lafay. Contribution expérimentale a l’aérodynamique dy cylindre. Revue Mécanique. 30, 431 ff. 1912.
are connected sequentially. As a result, the dynamo supplies current of any voltage, which can be regulated; the motor rotates with a frequency proportional to this voltage. Correspondingly, on the Bukau there are two motors for the independent rotation of both towers and three small dynamos—one for each tower and one for the ship’s general needs and for excitation of the magnetic field.
THE MAGNUS EFFECT AND THE WIND-POWERED SHIP
12) L. Prandtl. Article “Flüssigkeitbewegung” in “Handwörterbuch der Naturwissenschaften.” Jena. 1913.
13) H. Föttinger. Neue Grundlagen des Propellerproblems. Jahrb. d. Schiffbautechnik. Ges. 19, 426 ff. 1918.
14) L. Prandtl. Tragflügeltheorie. Nachr. d. Kgl. Ges. d. Wiss. Göttingen. 1908, p. 451 and 1909, p. 107.
15) A. Betz. Einführung in die Theorie der Flugzeugtragflügel. Naturwissenschaften, 1918, p. 557.
16) L. Prandtl. Tragflächenauftrieb und — Widerstand in der Theorie. Jahrb. d. Wiss. Ges. f. Luftfahrt. Berlin, 1920, p. 37 ff.
17) Th. v. Kármán. Über laminare und turbulente Reibung. Zeitschr. f. angew. Mathematik u. Mechanik. 1, 233, 1921.
18) A. Flettner. Die Anwendung der Erkenntnisse der Aerodynamik zum Windantrieb von Schiffen. Werft, Reederei, Hafen. 5, 657, 1924.
19) A. Betz. Der Magnuseffekt, die Grundlage der Flettnerrotors. ZS. d. Vereins deutsch. Ing. 69, 9, 1925.
20) J. Ackeret. Das Rotorschiff und seine physikalischen Grundlagen. Göttingen. 1925. Vandenhoeck u. Ruprecht-Verlag. Russian translation: Ackeret. The Rotor Ship. State Publishing House. Moscow. 1925.