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Autorotation of polygonal prisms with an upstream vane
JOURNALOF
Journal of Wind Engineering and Industrial Aerodynamics 73 (1998) 145 158
ELSEVIER
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Autorotation of polygonal prisms with an upstream vane B.W. Skews University of the Witwatersrand, PO WITS 2050, Johannesburg, South Africa Received 24 April 1996; accepted 9 July 1997
Abstract It has recently been shown that autorotation of bodies, with their axes normal to the airstream, is exhibited by bodies with polygonal sections having up to seven sides. In this work it is shown that the rotational speed can be enhanced, and prismatic bodies that do not autorotate on their own will do so, with the addition of a fixed upstream vane. Furthermore, it was found that the direction of rotation would change once for small vane gaps and three- and four-sided models as the vane angle is changed monotonically, but that three changes of direction were exhibited by bodies with five or more sides. For large vane gaps no changes in direction are exhibited, although for a five-sided body and intermediate gaps a case of two changes in direction was also recorded, and evidence of alternative final steady states arising from a given initial state was noted. Lift and drag coefficients for some of the cases are reported. © 1998 Elsevier Science B.V. All rights reserved.
Keywords: Autorotation; Magnus effect; Spinning bodies
I. Introduction Autorotation is the continuous rotation of a body in an airstream without the supply of external power. Bodies that autorotate are obvious candidates for devices which should be considered for the extraction of power from the wind. A comprehensive overview of the field of autorotation has been given by Lugt [1]. He distinguishes between cases where the axis of rotation is in the direction of the flow, as in the case of windmills, and those where the axis is normal to the flow, such as the Savonius rotor. In this latter case, there is a particular group of bodies which are not self starting and will only autorotate when given an initial impulse of a sufficiently large magnitude. A fiat plate pivoted on an axis of symmetry exhibits such behaviour. Such a body (an ordinary ruler, for example), if thrown whilst imparting a spin around its longitudinal axis will exhibit considerable lift due to the generation of Magnus forces resulting from the rotation, and will continue to spin as it falls to the ground. If such a plate is positioned in an airstream and externally powered, the variation of torque required 0167-6105/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 2 8 0 - 8
B . ~ Skews/J, Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
146
o ©
ANGULAR VELOCITY Fig. 1. Characteristic of an exlernally driven body which can autorotate.
with rotational speed is as shown in Fig. 1. At zero rotational speed the body takes up the position of maximum drag with the face of the plate normal to the flow. Any perturbation of the plate from this position results in the aerodynamic centre through which the net aerodynamic force, F, acts, to move off the pivot axis, resulting in a restoring torque bringing the body back to equilibrium. A positive mean torque is required to start, and keep the body rotating. As the speed increases a point will be reached where no external torque is required, and the line of action of the aerodynamic force again acts through the pivot, whereafter a negative torque (braking) is needed to keep the body rotating at a fixed speed. This first cross-over point is unstable. If the rotation is taken beyond this point and the external torque is removed the body will accelerate until the energy extracted exactly balances the dissipation, and autorotation at constant speed at position A will result. This is a stable flow condition since a slight increase in speed will result in the losses being larger than the aerodynamic torque derived from the wind and the body will slow back down to the equilibrium position. The rotation generates a lift force (Magnus force) and a body falling freely whilst autorotating will thus follow a trajectory angled to the ground. This factor is of importance in the distribution of debris following the breakup of vehicles in the air, the distribution of some seeds, and can also be used as a propulsive force, such as used on the Flettner rotor ship [2]. The flow mechanisms which are responsible for the features noted above have been elucidated by Lugt [3] for bodies of small thickness in relation to their chord. He presented numerical solutions of the two-dimensional Navier-Stokes equations for a plate of elliptical cross-section with a thickness to chord ratio of 0.1, rotating at constant angular velocity in a two-dimensional airstream. The main driving force generating the aerodynamic torque results from the suction due to a strong vortex shed from the retreating edge of the plate as indicated in Fig. 2 [4]. Stable autorotational conditions were shown to be satisfied for a ratio of plate tip speed to airspeed of
B.W Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
147
Fig. 2. Vortex development on the back face.
0.45. The fact that tipspeed is proportional to airspeed is to be expected from similarity considerations and is also borne out in experimental studies. Wind-tunnel tests using fiat plates of rectangular section, spanning the width of the test section and thus approximating two-dimensional flow have been reported by Skews [5], and show tipspeed ratios corresponding to the Lugt value. Furthermore, it was found that this value remains approximately constant over the full range of thickness ratios from thin plates to prisms of square cross section. This finding is in contrast to the results for finite aspect ratio plates. Iverson [4] correlated results of tests by Bustamante and Stone [6], Iversen [7], and Smith [8]. He examined effects of moment of inertia and showed that above a certain moment of inertia the effects are small and may be neglected. In the earlier flat-plate tests [5] moments of inertia were above this critical value and in the present tests with prismatic bodies they are very much higher still, and thus moment of inertia is not expected to be a factor during steady rotation conditions. In addition, Iverson's correlations showed that the tipspeed ratio is relatively independent of Reynold's number. His correlated data covers aspect ratios of between 0.25 and 4, and thickness ratios less than 0.5, and gives results somewhat different from the two-dimensional numerical and experimental studies; not only in that it predicts higher tipspeed ratios for thin plates but it also predicts that autorotation will not occur for square sections. The reasons for this have been discussed by Skews [5]. The finding that a square section body can autorotate raised the question of the autorotational properties of other bodies with other cross sections, such as regular polygons. Such experiments have been reported by Skews [9]. It is shown that triangular sections rotate the fastest but generate less lift than a fiat plate, for the same swept diameter. Four- and two-sided bodies spin with nearly the same tipspeed ratio but as the number of sides increases beyond four the tipspeed ratio reduces until eightsided bodies are used, where autorotation could not be induced. For all cases where autorotation occurred the lift force was found to be larger than that of a circular cylinder of the same diameter as that swept out by the prism and driven externally at the same rotational speed in an airstream of the same velocity I-2]. During these tests it was found that a very effective means of initiating rotation was to insert a thin fiat plate through a hole in the sidewall of the tunnel so as to direct the oncoming air
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B. gd Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
obliquely over one side of the model. It was noted that rapid acceleration of the cylinder occurred to rotational speeds some what higher than that during the main testing, and that there were multiple reversals of rotation under some circumstances, as well as hysteresis. This paper deals with a series of tests to explore these effects.
2. Apparatus The test equipment used was very similar to that described previously for the flat-plate tests [5], and is shown schematically in Fig. 3. The prism models were held between two conical pivots attached to a yoke straddling the wind tunnel and which was in turn connected to a simple two-component force balance. The pivots engage loosely in 1 m m diameter holes in the model. This mounting system was found to give very low friction. The models were made of a low-friction plastic, which is generally used as a bearing material, and ranged in size from 20 to 6 0 m m in diameter. (Diameter is defined as that of the circumscribing circle of the cross-sectional polygon.) The wind-tunnel cross section is 100 mm wide and 400 m m high and has a speed capability up to 50 m/s. It is of the suction type and is fitted with a smooth inlet contraction upstream of the transparent plastic test section, resulting in low freestream turbulence. Airspeed was measured with an upstream pitot tube on the tunnel centreline, and rotational speed with an electronic tachometer. Mean air density in the laboratory is about 0.97 kg/m 3. The vane consists of a smooth flat steel plate spanning the full width of the tunnel and with a chord of 50 m m (for most tests) and a thickness
DISK
F'ig. 3. Wind tunnel arrangement.
B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
149
of 1 mm. It is attached to the circular access doors on either side of the tunnel, and these doors are simply rotated to change the angle of inclination of the plate, over a range of 70 ° either side of the tunnel axis. Tests were only conducted with the plane of the vane passing through the axis of rotation of the model, as shown. The distance from the trailing edge of the plate to the model was generally held at 5 mm, except for some tests conducted to determine the influence of this gap. It was found that for a fixed wind-tunnel fan speed the velocity through the tunnel varied as the vane angle was changed as shown in Fig. 4, due to the change in blockage in the test section. Adjusting the tunnel speed for each vane angle would have resulted in very tedious testing. However, as will be shown in Section 3.1, similarity considerations allow this effect to be accounted for by suitable nondimensionalisation. Tests were conducted starting with the vane at one extreme setting (say + 70 °) and then moved in 5° steps to the other extreme. All measurements were made once the rotational speed of the body had settled to a new steady state. In view of both hysteresis and multiple changes of rotational direction being found under some circumstances, these adjustments in vane angle were made slowly, so as to minimise the influence of model inertia. This is particularly important in the vicinity of speed
Fig. 4. Sign convention employed.
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reversals. For some tests it is found that for a given starting vane angle the initial rotation of the body can be in either direction, whereas in others the flow over the vane results in forcing only one direction of rotation. In all figures the starting vane angle and sequence of tests will be indicated (generally starting at + 70 ° and reducing to - 70). The sign convention used is that shown in Fig. 5. With airflow from left to right, vane angles are measured with respect to the flow direction and are taken to be positive as shown. Clockwise rotation of the model is considered positive, as is the tipspeed for this direction of rotation. This convention results in lift forces arising from the Magnus effect to be positive upwards.
3. Results and discussion The first test undertaken was to establish if the tunnel speed affects the rotational behaviour. Fig. 6 gives the variation of rotational speed with vane angle for the tunnel velocity settings given in Fig. 4 above, for a 40 mm diameter triangular section body, with a 5 mm vane gap. Testing was started at + 70' vane angle which was then reduced. This factor is important because the curves do not have anti-symmetry about the origin, and so the direction of changes in vane angle need to be specified. It is noted that substantial speeds may be achieved. In the previous tests [4, 9] on both flat plates and bodies of polygonal section it was shown that the tipspeed of the model is directly proportional to the airspeed. This is also true in the present case with the vane, as is noted from Fig. 7. All the data of Fig. 6 collapse, to a significant degree, onto a single curve. Because of this, the approximately 15% variation in tunnel speed with vane angle noted above is not of concern in tipspeed ratio comparisons. Iverson [4] has shown that in order to account for bearing friction it is the slope of the tipspeed airspeed curve which should be used rather than the direct ratio of velocity. This correction has not been applied to the data in this present work since it has been 15000
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Fig. 6. Variation of spin with air speed and vane position.
B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
151
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Fig. 7. Non-dimensionalised rotational speed variation.
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shown previously [5] that the pivot design used in the present tests results in very small corrections, due to its low frictional torque. At zero vane angle the tipspeed ratio is similar (slightly lower) to that for the case with no vane [2]. This is a bit surprising in that the vane would be expected to inhibit the flow being swept around with the body, particularly considering the small gap (5 mm) between the vane and the body. Fig. 8 shows similar data for a 60 mm diameter triangular prism in a threedimensional representation, with the proportionality between rotational speed and airspeed being clear in the straight-line variation for each vane position. However, these results differ a bit from those for the 40 mm diameter triangular prism. In the region of 45 ° vane incidence the scatter increases and there is a noticeable dip in the curve. This dip appears to deepen as the air velocity decreases. It is evident from the comparison between Figs. 7 and 8 that the shape of the curve changes with diameter, and that there is lack of anti-symmetry to the phenomena even up to large vane angles, These issues will be addressed later.
152
Skews/'J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
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For a fixed vane gap of 5 mm, the results showing the effect of diameter and number of sides of the polygon are shown in the series of plots in Fig. 9a Fig. 9f, for tests at a nominal wind tunnel velocity of 35 m/s. In all cases, for large vane angles, the tipspeed ratios are smaller for the 20 mm diameter body. It should be recalled that the vane size remains constant for all tests and, thus, geometrical similarity is not maintained when model diameters are changed. This matter is addressed later. Flow visualisation would be useful to evaluate these effects, although the unsteadiness of the flow would make this difficult. The tipspeed ratios are generally slightly higher than the case with no vane. The general shape of the curves are, however, similar for all diameters for a given model profile.
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B.W. Skews/d. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
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Noticeable changes occur for the square body (Fig. 9b), compared to those of the triangular body discussed earlier. The dip noted for the triangular 60 mm diameter body is still evident but occurs at + 15 ° incidence and for both the 60 and 40 mm models. Tipspeed ratios for these two bodies are very similar, but the result for the 20 mm model changes markedly. Besides the tipspeed ratio reducing and the change in direction being delayed to a vane angle of about - 15 °, a substantial slowing down of the body occurs when e = - 30 °. For the five-sided body (Fig. 9c) even more marked changes are noted. The dip at ~ = + 15° deepens to the extent that the direction of rotation reverses, whereas the peak noted for the 20 mm cylinder square body, increases and is evident for all three models, resulting in a further two direction changes. The first change of direction now occurs for a substantial positive vane angle rather than the substantial negative angle of the three-sided body. Bodies of all three diameters also show three changes of direction as the vane angle changes from positive to negative. The results are again similar for the two larger diameters, with rotational speeds somewhat higher than that for the 20 mm body. With six sides (Fig. 9d), the pattern stays the same except that the regions of reversed rotation have broadened out and in the 20 ° case become approximately symmetrical about the origin. Attempts to get steady autorotation for the 2 0 m m diameter seven- and eight-sided bodies failed. The larger cylinders, however, continue to show the three changes in speed (Fig. 9e), although tipspeed ratios reduced significantly for the eight sided, 40 mm model, in a way similar to that which occurred previously with the 20 mm model. It could be that extending the tests to bodies with more sides would also result in the autorotation condition no longer being satisfied. However, it is noted that the eight-sided body still spins at a vane angle of zero, even though it could not be got to spin with the absence of the vane. The main features of the effects of the number of sides described above are summarised in Fig. 10 for the 40 mm diameter models. In view of these changes, the question that naturally arises is what the behaviour would be with different vane gaps. Fig. 1 la shows the results for a series of tests with
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the 40 mm diameter square section body with vane gaps of 5, 15 and 30 mm. The 5 mm gap test gives a curve very close to that previously obtained for the same boundary conditions (see Fig. 9b) with a single change of direction, giving an indication of the reproducibility of results. The result for the 15 and 30 mm gap are, however, completely different. Firstly, no reversal of rotation is exhibited, over the full range of vane angle, and secondly, the initial direction of spin can be arbitrarily chosen by the experimenter. Two tests for the 30 mm gap are shown, both starting with the maximum positive vane angle but with the initial rotational impulse in the opposite direction. The results are close to being mirror images of each other, in the vane angle axis. Similar symmetries would be obtained about the tipspeed ratio axis if the tests were started at negative vane angles. The 15 mm result does show a slowing down between 10" and -- 15', as if there is a tendency towards rotation reversal as occurs for the 5 mm case at these same angles. A similar test for the five-sided body showed that cases of two changes in direction may also be achieved. Fig. l l b shows the result of tests with vane gaps of 5 and 15 mm. The small-gap case behaves as described earlier with a change in direction before zero vane angle, followed by two further changes in the region of - 102 However, for the 15 mm case two flow cases are possible. The experimental results for the two cases are almost identical for reducing vane angles from + 80 to - 10, then in the one case the rotation jumps to a positive value and remains there, whereas in the other case it remains negative. There is clearly an unstable region in the vicinity of 10, where a slight bias in the flow one way or the other will result in subsequent
B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
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rotation being either positive or negative. Tests on this model with the 30 m m gap showed a single, or no change in direction, as for the four-sided case. It is to be expected that as the vane becomes more remote from the body so it will have reducing influence and the rotation will tend to remain in the direction in which it was initiated. Whilst it has clearly been shown above that the tipspeed ratio is independent of freestream velocity, the effect of the ratio of vane gap to body diameter is not clear. The results presented in Fig. 9 show that the curves for the 40 and 60 m m diameter models, in the cases of the four- to six-sided bodies approximately collapse onto each other, not withstanding the fact that the gap size remained at 5 ram, i.e. there is a lack of geometrical similarity. Thus, an additional series of tests were conducted with a vane of 75 m m chord, and a vane gap of 7.5 mm, together with the 60 m m diameter models. The geometry is thus the same as the 40 m m diameter tests with a 50 m m vane and 5 m m vane gap, but with a 50% increase in scale. The results for the three-, fiveand seven-sided bodies are given in Fig. 12. The results for the 60 m m tests with the smaller vane and gap are included for comparison. M a x i m u m tipspeed ratios are higher for the 60 m m tests than for the 40 m m geometrically similar tests thus indicating that true similarity is not achieved between the flows. It should be noted, however, that at the high vane angles at which these m a x i m u m tipspeed ratios are achieved, the tunnel blockage also increases by 50% and the tunnel height may also need to be scaled in order to achieve similarity. At the m a x i m u m vane angle of 70 ° the frontal area of the 60 m m diameter model together with the 75 m m vane is 44% of the tunnel test section area. 0.8 0.6
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156
B.I~ Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
It is interesting to note in the seven-sided tests that the tipspeed ratio is showing signs of decreasing at the high vane angles. It could be expected that as the vane angle gets towards 90 °, so a further unstable flow region may be reached with further changes in direction of rotation. The present experimental rig did not enable this to be tested. The fact that the body does not change direction at the position of zero vane angle will result in a hysteresis loop if the vane angle passes from the maximum to minimum position and then back again. Fig. 13 shows such a loop for the 60 m m diameter, triangular section model, with a 75 m m chord vane and a 7.5 m m vane gap. The vane angle was started at + 70 ~, and reduced in 5" steps to - 70 °, resulting in a change of direction of rotation just before 10 ~', followed by an increase in vane angle back to + 70 ~j, in this case resulting in the direction change just before + 10 °. The maximum tipspeed ratios that can be achieved with an upstream vane are significantly higher than without a vane (Fig. 14). This occurs for all models tested, having from three to eight-sides. No autorotation could be induced in an eight-sided body without a vane. Even with the vane set at zero incidence autorotation persists; for up to six sides at a value less than that with no vane, and at a higher value for seven and eight sides. Force measurements were conducted on the four- and five-sided bodies, with the 50 mm vane and a 5 m m vane gap. Fig. 15 shows the results. As expected the direction of the lift force correlates with the direction of rotation. Outside of the region where the direction of rotation changes ( - 20 < ~ < + 20) the lift coefficient varies nearly linearly with vane incidence. This is in the region where the rotational speed is nearly constant, and part of this lift must arise from the flow which has been deflected by the vane towards the body, resulting in a drag force which is inclined to the horizontal. Thus lift coefficients are considerably higher than for autorotation without the vane [9]. The drag coefficient shows an interesting h u m p in the region of zero vane angle. Without further information of the flow field patterns it is not easy to give a simple explanation for this effect.
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B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158 0.8
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4. Conclusions
The flow field about an autorotating prismatic body with an upstream vane is apparently very complex judging from the behaviour of the body, i.e. with multiple reversals of direction following monotonic changes in the vane direction. Explanations for these effects would appear to be best achieved through numerical modelling and time-resolved flow visualisation.
References [1] H.J. Lugt, Autorotation, Ann. Rev. Fluid Mech. 15 (1983) 123-147. [2] W.M. Swanson, The Magnus effect: a summary of investigations to date, J. Basic Eng. 83 (1961) 461-470. [3] H.J. Lugt, Autorotation of an elliptic cylinder about an axis perpendicular to the flow, J. Fluid Mech. 99 (1980) 817-840.
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[4] J.l). lversen, Autorotating flat-plate wings: the effect of the moment of inertia, geometry and Reynolds number, J. Fluid Mech. 92 (1979) 327 348. [5] B.W. Skews~ Autorotation of rectangular plates, J. Huid Mech. 217 (1990) 33 40. [6] A.G. Bustamante, G.W, Stone, The autorotation characteristics of various shapes for subsonic and hypersonic flows, AIAA Paper 69 (1969) 132. [7] JD. lversen, The Magnus rotor as an aerodynamic decelerator. Proc Aerodyn. Deceleration System Con[ vol. 2, Air Force Flight Test Center. Edwards AFB, CA, 1969, pp. 385 395. [8] E.H. Smith~ Autorotating wings: an experimental investigation, J. Fluid Mech. 50 (1971) 513 534. [9] B.W. Skews, A utorotation of many-sided bodies in an airstream, Nature 352 (1991) 512 513.
Journal of Wind Engineering and Industrial Aerodynamics 73 (1998) 145 158
ELSEVIER
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Autorotation of polygonal prisms with an upstream vane B.W. Skews University of the Witwatersrand, PO WITS 2050, Johannesburg, South Africa Received 24 April 1996; accepted 9 July 1997
Abstract It has recently been shown that autorotation of bodies, with their axes normal to the airstream, is exhibited by bodies with polygonal sections having up to seven sides. In this work it is shown that the rotational speed can be enhanced, and prismatic bodies that do not autorotate on their own will do so, with the addition of a fixed upstream vane. Furthermore, it was found that the direction of rotation would change once for small vane gaps and three- and four-sided models as the vane angle is changed monotonically, but that three changes of direction were exhibited by bodies with five or more sides. For large vane gaps no changes in direction are exhibited, although for a five-sided body and intermediate gaps a case of two changes in direction was also recorded, and evidence of alternative final steady states arising from a given initial state was noted. Lift and drag coefficients for some of the cases are reported. © 1998 Elsevier Science B.V. All rights reserved.
Keywords: Autorotation; Magnus effect; Spinning bodies
I. Introduction Autorotation is the continuous rotation of a body in an airstream without the supply of external power. Bodies that autorotate are obvious candidates for devices which should be considered for the extraction of power from the wind. A comprehensive overview of the field of autorotation has been given by Lugt [1]. He distinguishes between cases where the axis of rotation is in the direction of the flow, as in the case of windmills, and those where the axis is normal to the flow, such as the Savonius rotor. In this latter case, there is a particular group of bodies which are not self starting and will only autorotate when given an initial impulse of a sufficiently large magnitude. A fiat plate pivoted on an axis of symmetry exhibits such behaviour. Such a body (an ordinary ruler, for example), if thrown whilst imparting a spin around its longitudinal axis will exhibit considerable lift due to the generation of Magnus forces resulting from the rotation, and will continue to spin as it falls to the ground. If such a plate is positioned in an airstream and externally powered, the variation of torque required 0167-6105/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 2 8 0 - 8
B . ~ Skews/J, Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
146
o ©
ANGULAR VELOCITY Fig. 1. Characteristic of an exlernally driven body which can autorotate.
with rotational speed is as shown in Fig. 1. At zero rotational speed the body takes up the position of maximum drag with the face of the plate normal to the flow. Any perturbation of the plate from this position results in the aerodynamic centre through which the net aerodynamic force, F, acts, to move off the pivot axis, resulting in a restoring torque bringing the body back to equilibrium. A positive mean torque is required to start, and keep the body rotating. As the speed increases a point will be reached where no external torque is required, and the line of action of the aerodynamic force again acts through the pivot, whereafter a negative torque (braking) is needed to keep the body rotating at a fixed speed. This first cross-over point is unstable. If the rotation is taken beyond this point and the external torque is removed the body will accelerate until the energy extracted exactly balances the dissipation, and autorotation at constant speed at position A will result. This is a stable flow condition since a slight increase in speed will result in the losses being larger than the aerodynamic torque derived from the wind and the body will slow back down to the equilibrium position. The rotation generates a lift force (Magnus force) and a body falling freely whilst autorotating will thus follow a trajectory angled to the ground. This factor is of importance in the distribution of debris following the breakup of vehicles in the air, the distribution of some seeds, and can also be used as a propulsive force, such as used on the Flettner rotor ship [2]. The flow mechanisms which are responsible for the features noted above have been elucidated by Lugt [3] for bodies of small thickness in relation to their chord. He presented numerical solutions of the two-dimensional Navier-Stokes equations for a plate of elliptical cross-section with a thickness to chord ratio of 0.1, rotating at constant angular velocity in a two-dimensional airstream. The main driving force generating the aerodynamic torque results from the suction due to a strong vortex shed from the retreating edge of the plate as indicated in Fig. 2 [4]. Stable autorotational conditions were shown to be satisfied for a ratio of plate tip speed to airspeed of
B.W Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
147
Fig. 2. Vortex development on the back face.
0.45. The fact that tipspeed is proportional to airspeed is to be expected from similarity considerations and is also borne out in experimental studies. Wind-tunnel tests using fiat plates of rectangular section, spanning the width of the test section and thus approximating two-dimensional flow have been reported by Skews [5], and show tipspeed ratios corresponding to the Lugt value. Furthermore, it was found that this value remains approximately constant over the full range of thickness ratios from thin plates to prisms of square cross section. This finding is in contrast to the results for finite aspect ratio plates. Iverson [4] correlated results of tests by Bustamante and Stone [6], Iversen [7], and Smith [8]. He examined effects of moment of inertia and showed that above a certain moment of inertia the effects are small and may be neglected. In the earlier flat-plate tests [5] moments of inertia were above this critical value and in the present tests with prismatic bodies they are very much higher still, and thus moment of inertia is not expected to be a factor during steady rotation conditions. In addition, Iverson's correlations showed that the tipspeed ratio is relatively independent of Reynold's number. His correlated data covers aspect ratios of between 0.25 and 4, and thickness ratios less than 0.5, and gives results somewhat different from the two-dimensional numerical and experimental studies; not only in that it predicts higher tipspeed ratios for thin plates but it also predicts that autorotation will not occur for square sections. The reasons for this have been discussed by Skews [5]. The finding that a square section body can autorotate raised the question of the autorotational properties of other bodies with other cross sections, such as regular polygons. Such experiments have been reported by Skews [9]. It is shown that triangular sections rotate the fastest but generate less lift than a fiat plate, for the same swept diameter. Four- and two-sided bodies spin with nearly the same tipspeed ratio but as the number of sides increases beyond four the tipspeed ratio reduces until eightsided bodies are used, where autorotation could not be induced. For all cases where autorotation occurred the lift force was found to be larger than that of a circular cylinder of the same diameter as that swept out by the prism and driven externally at the same rotational speed in an airstream of the same velocity I-2]. During these tests it was found that a very effective means of initiating rotation was to insert a thin fiat plate through a hole in the sidewall of the tunnel so as to direct the oncoming air
148
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obliquely over one side of the model. It was noted that rapid acceleration of the cylinder occurred to rotational speeds some what higher than that during the main testing, and that there were multiple reversals of rotation under some circumstances, as well as hysteresis. This paper deals with a series of tests to explore these effects.
2. Apparatus The test equipment used was very similar to that described previously for the flat-plate tests [5], and is shown schematically in Fig. 3. The prism models were held between two conical pivots attached to a yoke straddling the wind tunnel and which was in turn connected to a simple two-component force balance. The pivots engage loosely in 1 m m diameter holes in the model. This mounting system was found to give very low friction. The models were made of a low-friction plastic, which is generally used as a bearing material, and ranged in size from 20 to 6 0 m m in diameter. (Diameter is defined as that of the circumscribing circle of the cross-sectional polygon.) The wind-tunnel cross section is 100 mm wide and 400 m m high and has a speed capability up to 50 m/s. It is of the suction type and is fitted with a smooth inlet contraction upstream of the transparent plastic test section, resulting in low freestream turbulence. Airspeed was measured with an upstream pitot tube on the tunnel centreline, and rotational speed with an electronic tachometer. Mean air density in the laboratory is about 0.97 kg/m 3. The vane consists of a smooth flat steel plate spanning the full width of the tunnel and with a chord of 50 m m (for most tests) and a thickness
DISK
F'ig. 3. Wind tunnel arrangement.
B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
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of 1 mm. It is attached to the circular access doors on either side of the tunnel, and these doors are simply rotated to change the angle of inclination of the plate, over a range of 70 ° either side of the tunnel axis. Tests were only conducted with the plane of the vane passing through the axis of rotation of the model, as shown. The distance from the trailing edge of the plate to the model was generally held at 5 mm, except for some tests conducted to determine the influence of this gap. It was found that for a fixed wind-tunnel fan speed the velocity through the tunnel varied as the vane angle was changed as shown in Fig. 4, due to the change in blockage in the test section. Adjusting the tunnel speed for each vane angle would have resulted in very tedious testing. However, as will be shown in Section 3.1, similarity considerations allow this effect to be accounted for by suitable nondimensionalisation. Tests were conducted starting with the vane at one extreme setting (say + 70 °) and then moved in 5° steps to the other extreme. All measurements were made once the rotational speed of the body had settled to a new steady state. In view of both hysteresis and multiple changes of rotational direction being found under some circumstances, these adjustments in vane angle were made slowly, so as to minimise the influence of model inertia. This is particularly important in the vicinity of speed
Fig. 4. Sign convention employed.
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reversals. For some tests it is found that for a given starting vane angle the initial rotation of the body can be in either direction, whereas in others the flow over the vane results in forcing only one direction of rotation. In all figures the starting vane angle and sequence of tests will be indicated (generally starting at + 70 ° and reducing to - 70). The sign convention used is that shown in Fig. 5. With airflow from left to right, vane angles are measured with respect to the flow direction and are taken to be positive as shown. Clockwise rotation of the model is considered positive, as is the tipspeed for this direction of rotation. This convention results in lift forces arising from the Magnus effect to be positive upwards.
3. Results and discussion The first test undertaken was to establish if the tunnel speed affects the rotational behaviour. Fig. 6 gives the variation of rotational speed with vane angle for the tunnel velocity settings given in Fig. 4 above, for a 40 mm diameter triangular section body, with a 5 mm vane gap. Testing was started at + 70' vane angle which was then reduced. This factor is important because the curves do not have anti-symmetry about the origin, and so the direction of changes in vane angle need to be specified. It is noted that substantial speeds may be achieved. In the previous tests [4, 9] on both flat plates and bodies of polygonal section it was shown that the tipspeed of the model is directly proportional to the airspeed. This is also true in the present case with the vane, as is noted from Fig. 7. All the data of Fig. 6 collapse, to a significant degree, onto a single curve. Because of this, the approximately 15% variation in tunnel speed with vane angle noted above is not of concern in tipspeed ratio comparisons. Iverson [4] has shown that in order to account for bearing friction it is the slope of the tipspeed airspeed curve which should be used rather than the direct ratio of velocity. This correction has not been applied to the data in this present work since it has been 15000
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Fig. 6. Variation of spin with air speed and vane position.
B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
151
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Fig. 7. Non-dimensionalised rotational speed variation.
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shown previously [5] that the pivot design used in the present tests results in very small corrections, due to its low frictional torque. At zero vane angle the tipspeed ratio is similar (slightly lower) to that for the case with no vane [2]. This is a bit surprising in that the vane would be expected to inhibit the flow being swept around with the body, particularly considering the small gap (5 mm) between the vane and the body. Fig. 8 shows similar data for a 60 mm diameter triangular prism in a threedimensional representation, with the proportionality between rotational speed and airspeed being clear in the straight-line variation for each vane position. However, these results differ a bit from those for the 40 mm diameter triangular prism. In the region of 45 ° vane incidence the scatter increases and there is a noticeable dip in the curve. This dip appears to deepen as the air velocity decreases. It is evident from the comparison between Figs. 7 and 8 that the shape of the curve changes with diameter, and that there is lack of anti-symmetry to the phenomena even up to large vane angles, These issues will be addressed later.
152
Skews/'J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
B.,q"~
For a fixed vane gap of 5 mm, the results showing the effect of diameter and number of sides of the polygon are shown in the series of plots in Fig. 9a Fig. 9f, for tests at a nominal wind tunnel velocity of 35 m/s. In all cases, for large vane angles, the tipspeed ratios are smaller for the 20 mm diameter body. It should be recalled that the vane size remains constant for all tests and, thus, geometrical similarity is not maintained when model diameters are changed. This matter is addressed later. Flow visualisation would be useful to evaluate these effects, although the unsteadiness of the flow would make this difficult. The tipspeed ratios are generally slightly higher than the case with no vane. The general shape of the curves are, however, similar for all diameters for a given model profile.
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B.W. Skews/d. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
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Noticeable changes occur for the square body (Fig. 9b), compared to those of the triangular body discussed earlier. The dip noted for the triangular 60 mm diameter body is still evident but occurs at + 15 ° incidence and for both the 60 and 40 mm models. Tipspeed ratios for these two bodies are very similar, but the result for the 20 mm model changes markedly. Besides the tipspeed ratio reducing and the change in direction being delayed to a vane angle of about - 15 °, a substantial slowing down of the body occurs when e = - 30 °. For the five-sided body (Fig. 9c) even more marked changes are noted. The dip at ~ = + 15° deepens to the extent that the direction of rotation reverses, whereas the peak noted for the 20 mm cylinder square body, increases and is evident for all three models, resulting in a further two direction changes. The first change of direction now occurs for a substantial positive vane angle rather than the substantial negative angle of the three-sided body. Bodies of all three diameters also show three changes of direction as the vane angle changes from positive to negative. The results are again similar for the two larger diameters, with rotational speeds somewhat higher than that for the 20 mm body. With six sides (Fig. 9d), the pattern stays the same except that the regions of reversed rotation have broadened out and in the 20 ° case become approximately symmetrical about the origin. Attempts to get steady autorotation for the 2 0 m m diameter seven- and eight-sided bodies failed. The larger cylinders, however, continue to show the three changes in speed (Fig. 9e), although tipspeed ratios reduced significantly for the eight sided, 40 mm model, in a way similar to that which occurred previously with the 20 mm model. It could be that extending the tests to bodies with more sides would also result in the autorotation condition no longer being satisfied. However, it is noted that the eight-sided body still spins at a vane angle of zero, even though it could not be got to spin with the absence of the vane. The main features of the effects of the number of sides described above are summarised in Fig. 10 for the 40 mm diameter models. In view of these changes, the question that naturally arises is what the behaviour would be with different vane gaps. Fig. 1 la shows the results for a series of tests with
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Fig. 10. Effect of number of sides on the speed ratio.
154
B.~ Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158 o.8
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the 40 mm diameter square section body with vane gaps of 5, 15 and 30 mm. The 5 mm gap test gives a curve very close to that previously obtained for the same boundary conditions (see Fig. 9b) with a single change of direction, giving an indication of the reproducibility of results. The result for the 15 and 30 mm gap are, however, completely different. Firstly, no reversal of rotation is exhibited, over the full range of vane angle, and secondly, the initial direction of spin can be arbitrarily chosen by the experimenter. Two tests for the 30 mm gap are shown, both starting with the maximum positive vane angle but with the initial rotational impulse in the opposite direction. The results are close to being mirror images of each other, in the vane angle axis. Similar symmetries would be obtained about the tipspeed ratio axis if the tests were started at negative vane angles. The 15 mm result does show a slowing down between 10" and -- 15', as if there is a tendency towards rotation reversal as occurs for the 5 mm case at these same angles. A similar test for the five-sided body showed that cases of two changes in direction may also be achieved. Fig. l l b shows the result of tests with vane gaps of 5 and 15 mm. The small-gap case behaves as described earlier with a change in direction before zero vane angle, followed by two further changes in the region of - 102 However, for the 15 mm case two flow cases are possible. The experimental results for the two cases are almost identical for reducing vane angles from + 80 to - 10, then in the one case the rotation jumps to a positive value and remains there, whereas in the other case it remains negative. There is clearly an unstable region in the vicinity of 10, where a slight bias in the flow one way or the other will result in subsequent
B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158
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rotation being either positive or negative. Tests on this model with the 30 m m gap showed a single, or no change in direction, as for the four-sided case. It is to be expected that as the vane becomes more remote from the body so it will have reducing influence and the rotation will tend to remain in the direction in which it was initiated. Whilst it has clearly been shown above that the tipspeed ratio is independent of freestream velocity, the effect of the ratio of vane gap to body diameter is not clear. The results presented in Fig. 9 show that the curves for the 40 and 60 m m diameter models, in the cases of the four- to six-sided bodies approximately collapse onto each other, not withstanding the fact that the gap size remained at 5 ram, i.e. there is a lack of geometrical similarity. Thus, an additional series of tests were conducted with a vane of 75 m m chord, and a vane gap of 7.5 mm, together with the 60 m m diameter models. The geometry is thus the same as the 40 m m diameter tests with a 50 m m vane and 5 m m vane gap, but with a 50% increase in scale. The results for the three-, fiveand seven-sided bodies are given in Fig. 12. The results for the 60 m m tests with the smaller vane and gap are included for comparison. M a x i m u m tipspeed ratios are higher for the 60 m m tests than for the 40 m m geometrically similar tests thus indicating that true similarity is not achieved between the flows. It should be noted, however, that at the high vane angles at which these m a x i m u m tipspeed ratios are achieved, the tunnel blockage also increases by 50% and the tunnel height may also need to be scaled in order to achieve similarity. At the m a x i m u m vane angle of 70 ° the frontal area of the 60 m m diameter model together with the 75 m m vane is 44% of the tunnel test section area. 0.8 0.6
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156
B.I~ Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145 158
It is interesting to note in the seven-sided tests that the tipspeed ratio is showing signs of decreasing at the high vane angles. It could be expected that as the vane angle gets towards 90 °, so a further unstable flow region may be reached with further changes in direction of rotation. The present experimental rig did not enable this to be tested. The fact that the body does not change direction at the position of zero vane angle will result in a hysteresis loop if the vane angle passes from the maximum to minimum position and then back again. Fig. 13 shows such a loop for the 60 m m diameter, triangular section model, with a 75 m m chord vane and a 7.5 m m vane gap. The vane angle was started at + 70 ~, and reduced in 5" steps to - 70 °, resulting in a change of direction of rotation just before 10 ~', followed by an increase in vane angle back to + 70 ~j, in this case resulting in the direction change just before + 10 °. The maximum tipspeed ratios that can be achieved with an upstream vane are significantly higher than without a vane (Fig. 14). This occurs for all models tested, having from three to eight-sides. No autorotation could be induced in an eight-sided body without a vane. Even with the vane set at zero incidence autorotation persists; for up to six sides at a value less than that with no vane, and at a higher value for seven and eight sides. Force measurements were conducted on the four- and five-sided bodies, with the 50 mm vane and a 5 m m vane gap. Fig. 15 shows the results. As expected the direction of the lift force correlates with the direction of rotation. Outside of the region where the direction of rotation changes ( - 20 < ~ < + 20) the lift coefficient varies nearly linearly with vane incidence. This is in the region where the rotational speed is nearly constant, and part of this lift must arise from the flow which has been deflected by the vane towards the body, resulting in a drag force which is inclined to the horizontal. Thus lift coefficients are considerably higher than for autorotation without the vane [9]. The drag coefficient shows an interesting h u m p in the region of zero vane angle. Without further information of the flow field patterns it is not easy to give a simple explanation for this effect.
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B.W. Skews/J. Wind Eng. Ind. Aerodyn. 73 (1998) 145-158 0.8
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4. Conclusions
The flow field about an autorotating prismatic body with an upstream vane is apparently very complex judging from the behaviour of the body, i.e. with multiple reversals of direction following monotonic changes in the vane direction. Explanations for these effects would appear to be best achieved through numerical modelling and time-resolved flow visualisation.
References [1] H.J. Lugt, Autorotation, Ann. Rev. Fluid Mech. 15 (1983) 123-147. [2] W.M. Swanson, The Magnus effect: a summary of investigations to date, J. Basic Eng. 83 (1961) 461-470. [3] H.J. Lugt, Autorotation of an elliptic cylinder about an axis perpendicular to the flow, J. Fluid Mech. 99 (1980) 817-840.
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[4] J.l). lversen, Autorotating flat-plate wings: the effect of the moment of inertia, geometry and Reynolds number, J. Fluid Mech. 92 (1979) 327 348. [5] B.W. Skews~ Autorotation of rectangular plates, J. Huid Mech. 217 (1990) 33 40. [6] A.G. Bustamante, G.W, Stone, The autorotation characteristics of various shapes for subsonic and hypersonic flows, AIAA Paper 69 (1969) 132. [7] JD. lversen, The Magnus rotor as an aerodynamic decelerator. Proc Aerodyn. Deceleration System Con[ vol. 2, Air Force Flight Test Center. Edwards AFB, CA, 1969, pp. 385 395. [8] E.H. Smith~ Autorotating wings: an experimental investigation, J. Fluid Mech. 50 (1971) 513 534. [9] B.W. Skews, A utorotation of many-sided bodies in an airstream, Nature 352 (1991) 512 513.