# Rotor lift and drag

The lift forces of a spinning cylinder are very much higher than those of a textile sail or an aircraft wing having the same projected area. Potential theory predicts that the lift per unit length of rotor should be 2n times the product of the surface speed of the rotor and far-field wind speed. This means that, for a constant rotor speed, it will rise with the first power of wind speed rather than with the square. If rotor surface speed and wind speed are kept in proportion, square law equations can be used (as in aircraft design) for comparison with wings and sails. The spin ratio (defined as local rotor speed over far-field wind speed in a frame moving with the vessel) acts such as the angle of incidence of the aerofoil section of an aircraft wing. The lift coefficient from ideal potential theory, as used with the square of velocity in aircraft design, is shown as the heavy line of Figure 11.6. The open circles are from wind tunnel tests by Reid (1924) on a 115-mm diameter cylinder. He reported excessive vibration sufficient to stop the test at 3000 rpm and a tunnel speed of 10 ms-1, which would be a spin ratio of 1.79, but did not report results for spin ratios above 4.32.

Figure 11.7 John Marples' Cloudia (a rebuilt Searunner 34), with Thom fences on test at Fort Pierce, FL, February 2008. With a rotor drive power of 600 W, she could sail faster than the beam wind, stop, go into reverse and yaw 180° in either direction about her own axis. Funding for work on Cloudia was provided by the Discovery Channel and organized by Impossible Pictures. (© Discovery Channel.)

Figure 11.7 John Marples' Cloudia (a rebuilt Searunner 34), with Thom fences on test at Fort Pierce, FL, February 2008. With a rotor drive power of 600 W, she could sail faster than the beam wind, stop, go into reverse and yaw 180° in either direction about her own axis. Funding for work on Cloudia was provided by the Discovery Channel and organized by Impossible Pictures. (© Discovery Channel.)

It is well known that part of the drag on an aircraft wing is due to the permanent tip vortex generated by the positive pressure on the under surface driving air to the negative pressure on the upper surface. The effect can be minimized by high aspect-ratio wings, such as those of the albatross, and by tip fins. It was for this reason that Flettner added discs to the tops of his rotors (see Figure 11.5). As a further design development, Thom (1934) experimented with multiple discs (or fences) and found that they produced very much higher lift coefficients and sometimes even negative drag coefficients. His data for disc diameters three times the rotor diameter placed at intervals of 0.75 of the rotor diameter are plotted as open triangles in Figure 11.6.

The negative drag coefficients imply that some forward drive power is being taken from the rotor drive. Also plotted in Figure 11.6 with filled circles are coefficients from a numerical simulation carried out by Mittal & Kumar (2003) for an infinitely long cylinder. The falling drag values, even going negative, are of interest and provide qualitative support for Thom's observations. All predictions agree quite well up to spin ratios of approximately 3, but diverge for higher values. A photograph of a sea-going yacht conversion by John Marples incorporating Flettner rotors with Thom fences is shown in Figure 11.7. An artist's impression of the final spray vessel is shown in Figure 11.8.

Figure 11.8 A conceptual Flettner spray vessel with Thom fences. The wind would be blowing from the reader's right-hand side, the rotor spin would be clockwise seen from above and rotor thrust to the left. Vessels can also report sea and air temperatures, humidity, solar input, the direction and velocities of winds and currents, atmospheric pressure, visibility, cloud cover, plankton count, aerosol count, salinity and radio reception and could even rescue yachtsmen in distress. (© J. MacNeill 2006.)

Figure 11.8 A conceptual Flettner spray vessel with Thom fences. The wind would be blowing from the reader's right-hand side, the rotor spin would be clockwise seen from above and rotor thrust to the left. Vessels can also report sea and air temperatures, humidity, solar input, the direction and velocities of winds and currents, atmospheric pressure, visibility, cloud cover, plankton count, aerosol count, salinity and radio reception and could even rescue yachtsmen in distress. (© J. MacNeill 2006.)

Wind tunnel balances from the pre-war years had none of the force-sensing transducers allowed by later electronics. It must have been difficult to make accurate small drag-force measurements on vortex shedding rotors that were being fed with mechanical power. But if Mittal and Thom are right, we can design some very exciting sailing ships.

Figure 11.9, also taken from Mittal & Kumar, shows lift coefficients against time after spin-up for a series of spin ratios from 0 to 5. There is an interesting build-up of vibrations for spin ratios up to 2 and also between 4 and 4.8 which are reminiscent of vortex shedding and are in good agreement with Reid's reported vibrations. (As a cautionary note, however, the Reynolds number in Mittal & Kumar's simulations is only 200, so there must be some doubt as to whether the same features will be present in a full-scale craft where the Reynolds numbers will be at least four orders of magnitude greater.)

In a subsequent paper, Mittal (2004) shows span-wise axial oscillations in air velocity at various times after spin-up. The pitch of the oscillations is close to that of the fences suggested by Thom and it may be that the superior performance of fenced rotors is caused by the suppression of this instability. Thom measured the

Figure 11.9 Numerical lift coefficients versus time (Mittal & Kumar 2003). (Courtesy of Journal of Fluid Mechanics.)

torque needed to spin his cylinders, but then made a mistake in scaling up the torque coefficient to the much higher Reynolds numbers needed for practical applications. This was spotted by Norwood (1991), who confirmed his own torque calculations with a model test.

The patterns of the air flow associated with the vibrations of Mittal's numerical predictions show clear vortex shedding as in the flow over a stalled aircraft wing. The vortex axes are parallel to the spin axis. Some aircraft designers put small vortex generators on the upper wing surfaces to induce pairs of vortices with axes parallel to the line of flight. These stabilize the air flow against separation. A single disc will centrifuge air outwards and lose all its kinetic energy. But perhaps closely packed discs with root fillets, as in Figure 11.8, may also induce pairs of vortices returning some of the kinetic energy of spin to the rotor core. The resistance to buckling of the double curvature of the fillets on the discs will make them much stronger.