1. Introduction
Radio-control model glider pilots have found a new way of dynamic soaring in the lee of a hill. Normally, soaring is done on the windward side of a hill to take advantage of the up-draughts as the wind blows up and over the hill; the lee side of the hill is normally avoided because of down-draughts. However, it has been found that it is possible to maintain height and position in a circular flight path and achieve huge speeds, on the lee-side of a hill, where there is a marked shear boundary between the wind blowing over the top of the hill and a relatively still area below. This has been recorded by hand-held radar speed detectors with speeds of more than 500mph!
Figure 1
2. Rayleigh cycle
Again, the Rayleigh cycle of dynamic soaring is cited as the mechanism in use. It goes something like this: the model initially flies in the still air below the shear boundary and climbs upwind through the shear boundary into the fast moving air; ground speed is preserved and airspeed increases. The glider turns downwind and descends through the shear boundary; ground speed is preserved and airspeed increases. The circuit is completed in still air below the shear boundary with a little loss of speed and the cycle is repeated.
So what is wrong with that? In this form of dynamic soaring both airspeed and ground-speed must increase but the theory says that airspeed will increase only if actual speed (ground-speed) is conserved. If actual speed reduces, the airspeed will not increase by the amount of the wind shear. The theory does not explain why the actual speed increases during the downwind turn. Also, the wind gradient has maximum effect only if the glider penetrates the shear boundary on upwind or downwind headings. If the glider penetrates the shear boundary on approximately cross wind headings or in a turn, the headwind component and the effect of the wind-gradient is much reduced.
In order for both airspeed and ground-speed to increase there has to be a force acting in the direction of motion to achieve that acceleration and also to overcome the increasing drag. The wind gradient theory does not show such a force. The wind is an obvious source of energy but how is energy transferred from the wind to the glider? Wind-gradients clearly play a part in RC dynamic soaring but there is more to it than that. There is something wrong with the theory.
This is also not the same as regular albatross dynamic soaring. As explained elsewhere in this website, in albatross dynamic soaring airspeed can be maintained in a windward turn; but in the case of RC dynamic soaring the ‘windward’ turn is made in still air with reducing airspeed and ground-speed and there is no gain of energy.
This form of RC dynamic soaring is not the same as what the albatrosses do and nor is it completely explained by the Rayleigh cycle. I think there are two additional mechanisms involved. One is like a kind of auto-rotation, the kind of thing that keeps the rotor of an auto-gyro or the vanes of a vertical-axis windmill turning. The other mechanism is a high-G variation of what the albatrosses do.
3. The Kick
RC dynamic soaring or Lee soaring can be partly explained by an effect similar to the auto-rotation which drives a windmill or the rotor of an auto-gyro or a helicopter in a power-off glide. Such mechanisms have axles or rotor hubs to constrain the motion of the rotor and ensure a permanent high angle of attack. However, the flight path of the RC glider is controlled by the pilot and can only sustain high angles of attack briefly before high drag and the stability of the machine reduce the effect, so that the effect is only transitory.
Figure 3
This is how the auto-rotation mechanism works: see the diagram above which is a plan view of the glider circling to the right with a steep angle of bank. At position 1, below the shear boundary in still air, the glider has a small angle of attack and the lift L and drag D forces are resolved to give resultant R.
At position 2, at the cross-wind position, the glider penetrates the shear boundary briefly. The glider continues its original curved path G over the ground but the wind W suddenly appears from the left and causes a change in the direction of the air-velocity V and a slight increase in air-speed. The difference in the direction of the ground-velocity and the air-velocity is essentially an angle of drift but in this case acts as an angle of attack. The air-velocity and the angle of drift/attack is caused by the vector addition of wind-velocity and ground-velocity rather than ground-velocity being the vector addition of wind-velocity and air-velocity.
The sudden increase in angle of attack increases lift and drag and also the load-factor (G-loading) but, more importantly, the lift-drag resultant R tilts forward, giving the glider a kick, like a pulse of thrust T, in the direction of its ground-velocity G. Force component C acts as the centripetal force to keep the turn going. The pulse of thrust, the kick, increases the ground-speed and the airspeed. There is a further slight increase of airspeed as soon as the glider drops below the shear boundary, the circuit being completed in the still air below the shear boundary with a small loss of airspeed. Thus both ground-speed and air-speed increase on every circuit.
It is similar to a kind of vertical axis wind turbine or a cup anemometer. The wind-turbine has a vertical axis and three arms, but with a vertical airfoil blade at the end of each arm instead of the cup. Like the cup anemometer, this device will also rotate regardless of the wind direction. Just imagine instead of three arms, a single arm and single airfoil and there is your glider in a steep turn sipping at the wind at the upwind peak of the orbit, with the RC pilot maintaining the circle.
4. Adapting the Windward Turn Theory to RC dynamic soaring
The ‘kick’ or surge is a consequence of flying through a sudden, shallow wind shear giving a brief impulse. What happens when an RC glider circles, climbing and descending in a deeper wind-gradient; bearing in mind that the pilot cannot see where the wind-gradient or the shear boundary begins and ends? I have to admit that I thought the kick was all there was to it. I am slightly surprised to get a positive result by applying the Windward Turn Theory to RC dynamic soaring. The Windward Turn Theory explains how albatrosses are able to dynamic soar but I must emphasise that the same basic mechanism works differently in the two cases because the albatrosses fly a low-G, undulating flight path and the RC gliders fly a high-G circular flight path.
Figure 4
In this section, the Windward Turn Theory is adapted to simulate RC dynamic soaring. The math is the same as described in the Analysis section. All of the diagrams in this section are from an Excel spread-sheet using a single set of equations and starting data. The equations are the same as used to plot the illustration of the albatross undulating flight path, in the analysis section, only the shape of the manoeuvre is different. The spreadsheet calculates and plots one 360 degree circle at 10 degree intervals of wind-angle, starting at the lowest point, on a crosswind heading (270o) in the middle of the windward turn. It uses airspeed, angle of bank and angle of climb, load factor, mass, lift/drag ratio, the wind and drift angle and the aerodynamic forces and gravity. The results in numerical or graphical form, can be inspected to see how much speed or height is gained. If airspeed is gained, that value is manually entered as the starting airspeed and the next circle is plotted.
And this is the point: I am not simply assuming that airspeed gained in the wind-gradient equals airspeed lost due to drag. I am applying the aerodynamic and gravitational forces together with the rate of change of the headwind component H and discovering whether or not there is an increase of speed. This is intended to show that the manoeuvre is not energy-neutral, as some people think, where airspeed gained in the wind-shear equals airspeed lost due to drag. Rather it shows that there is a net energy gain in the form of increased airspeed and ground-speed. The energy is derived from the forces acting on the aircraft. The forces are derived from momentum given to the air.
Figure 5 a & b
5. Aerodynamic force.
The glider flies continuous circles with a steep angle of bank and a high load-factor. The next diagram shows a plan view of the triangle of velocity and the force components at one wind-angle in the leeward turn. The glider is turning right, the wind is from the left. The left hand diagram (a) shows the triangle of velocity and the force components relative to the air-velocity. V is air-velocity, W is wind-velocity and G is ground-velocity. FC is the centripetal component of the lift force (the vertical component of lift is not shown). FD is the drag force. FR is the horizontal resultant, the vector sum of FC and FD. Angle d is the drift-angle (on the same side as the angle-of-bank, both to the right). Angle y is the wind-angle.
The right-hand diagram (b) shows the same position but shows the force components relative to the ground-velocity. K is a component of the ground-velocity. H is the headwind component. V is the sum of K and H. FR is the horizontal resultant (same as diagram a). FGC is the centripetal component giving the curvature of the ground track; FGT is the tangential component which gives the acceleration of ground-speed; in this case it is propulsive and explains why the ground-speed increases when the glider turns downwind.
As the glider turns right, component K increases and headwind component H reduces. The acceleration of air-speed is the sum of the rate of change of K and the rate of change of H. The ground-speed will increase because of force component FGT. If FGT is great enough so that the rate of increase of K is greater than the rate of reduction of H, then V will increase. This is what happens in RC dynamic soaring, giving an increase of both V and G. The reason is that the glider is flown with a steep angle of bank and very high load-factor, enabling a very large value of FR and of FGT and FGC. This is possible because the glider is built extremely strong and remote controlled and able to resist the very high G-loading.
This is different to albatross dynamic soaring in which high G-loading is avoided because it will lead to a high energy expenditure on the part of the bird. Albatross dynamic soaring is done at approximately one-G all the time.
Figure 5 c & d
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The next diagram shows all the same parameters but in the windward turn, the glider having completed half a circle. The difference between the windward and leeward turns is that the wind-velocity is much reduced due to the wind-gradient and the angle of bank is on the opposite side to the angle of drift. Angle of bank is still to the aircraft right and drift to the aircraft left.
The force component FGT is now retarding, reducing the ground speed as the glider turns upwind. Compared to cross-wind albatross dynamic soaring, in Rc dynamic soaring the wind-gradient is much more important because the whole circle is flown.
6. Momentum
Figure 6 a & b
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Aerodynamic forces are the equal and opposite reaction to the rate of change of momentum given to the air through which the aircraft is passing. In a turn, the lift force tilts with the aircraft bank-angle and the horizontal component acts as a centripetal force which causes the aircraft to turn.
In the leeward turn, part of the momentum given to the air is horizontal, opposite to the wind-velocity. (Fig 6a) Therefore, as the aircraft turns downwind, the wind loses momentum as the aircraft gains momentum.
In the windward turn (Fig 6b) the aircraft loses momentum and the wind gains momentum but the aircraft loses less momentum than it gained in the leeward turn and comes out with more speed overall.
This is because, in the leeward turn at the top of the wind-gradient, the angle of drift is greater compared with the smaller angle of drift in the windward turn at the bottom of the wind-gradient where the wind is less. This means that the effect of the aerodynamic forces on the tangential acceleration of the aircraft are greater in the leeward turn than in the windward turn, and therefore the exchange of momentum is biased in favour of the aircraft.
This exchange of momentum and energy between the aircraft and the wind is essentially the same as with albatross dynamic soaring but with different combinations of angle of bank, load-factor, rate of turn and drift.
Figure 7a
7. The Wind gradient
The wind encountered by the aircraft in the spread-sheet, depends on the aircraft height within the wind-gradient layer, starting at the lowest point of the circle and climbing and descending through the wind gradient. To generate a wind profile in the spread-sheet, the wind used is the product of the maximum wind and the logarithm base 10 of the aircraft height which gives a wind-versus-height profile as seen in the diagram. (Fig 7a) The imaginary hill-top producing this imagined effect is some arbitrary height and distance upwind. The height gained by the aircraft will depend on the angle of climb and the airspeed. However, the angle of climb is assumed to be small, approximately 5 degrees, to avoid introducing extra geometric complications.
Figure 7b
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In this example the height of the aircraft during the circle is shown in this diagram. (Fig 7b) It starts at an arbitrary 1m and peaks at just over 3m. It ends with a slight height excess. Avoiding negative heights makes operation of the spreadsheet easier and is, of course, a necessity to avoid losing energy to gravity. If the maximum wind speed is increased and the circle is allowed to drift downwind of its starting position, it will generate negative heights and this is avoided by increasing the bank-angle as seen in the ground-plot.
8. Angle of bank and Load Factor
Figure 8a
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In real life the angle of bank (and the load-factor) is under the control of the pilot and must be varied to keep the circular flight path of the aircraft in approximately the same location over the ground, while the wind is trying to blow the aircraft downwind. In the simulation, the aircraft angle of bank is generated by a sine function depending on the wind-angle, that is the position of the aircraft in the circle, and on the bank-angle amplitude. (Fig 8a). The bank angle amplitude is the difference between the maximum and minimum bank angles. In this diagram the bank angle varies from 32 to 68 degrees.
Figure 8b
A phase angle is added to the wind-angle to position the maximum angle of bank approximately (just before) the 180 degree wind-angle which is the downwind heading. The least angle of bank is then at the zero or 360 degree upwind wind-angle. The load-factor can be factored to a greater value than that in level flight if required.
Each time a change is made to any of the input parameters, the ground plot is inspected and the bank angle adjusted, to ensure the aircraft ends up approximately back at its ground starting position to ensure it does not drift downwind. The diagrams of bank and load-factor show the variation of bank from 32 to 69 degrees and of load-factor from 1.9 to 5G. The asymmetry of this diagram is caused by the horizontal axis being intervals of wind-angle and not equal time. (Fig 8b)
This is a mild example flown at relatively low speed. However, the manoeuvre produces a gain of airspeed with every circle. Therefore, if the exercise is repeated, the airspeed rapidly builds up. In this case, when the greater starting airspeed is entered, different values of lift/drag ratio have to be used to simulate the increasing drag load.
9. Ground-plot and Air-plot
Figure 9a
The next diagram, figure 9a, shows the Ground-plot on a horizontal grid of 10 by 10 meter squares, starting at the zero coordinates, with a wind-angle (heading) of 270 degrees, turning right. The wind is from the North. The diameter of the circle is 50 to 60m and ends up about 5m West and slightly downwind of the starting point. To achieve this result, the bank-angle is least on the upwind heading on the left side of the diagram and greatest on the downwind heading on the right side of the diagram. (See fig 8a)
Figure 9b
The Air-plot, figure 9b, is the path of the aircraft through the air, to which the wind is added to give the Ground-plot, starting at the zero point on a heading of 270 degrees turning right. It illustrates the increased rate of turn and reduced turn radius caused by the increased angle of bank, during the second half of the circle. This is modified by the wind as seen in the ground plot.
In effect, the path through the air is moving upwind; which gives a clue as to how albatrosses are able to dynamic soar upwind. (See the Upwind dynamic soaring page)
10. Airspeed and ground-speed
The speed diagram, figure 10, shows the airspeed and ground-speed during one circle. The airspeed is a nominal 20 m/s at the start, on the 270 degree crosswind heading, at the lowest point of the wind-gradient where the wind-speed is zero and therefore the drift angle is zero and the ground speed is the same as the airspeed.
The ground-speed reduces in the first quarter from 270 to 360, as you would expect in a windward turn and then increases during the downwind or leeward turn from 360 to 180, the middle half of the diagram. During the fourth quarter from 180 to 270, the ground-speed starts to reduce again but ends up about 2 m/s more than the starting ground-speed. At the end, the air-speed and ground-speed come back together as the drift angle reduces to zero. It clearly shows a gain of speed overall.
This new airspeed can be used as the starting point for the next circle, although the other parameters will need to be changed especially the assumed lift/drag ratio and the angles of bank.
Figure 10
The airspeed is affected by both the acceleration of the ground speed and by the rate of change of the headwind component. The rate of change of the headwind component is affected by both the rate of turn and by the wind-gradient. These effects result in large variations of the ground-speed and smaller but complementary variations of the airspeed. During the first and fourth quarters, making up the windward turn, the wind-gradient makes the headwind component reduce. However, the rate of turn causes the headwind component to increase and reduces airspeed losses (Rh, the rate of change of the headwind component is slightly negative. See the Analysis section for an explanation of Rh and Rk).
During the the leeward turn, the middle half of the circle from 360 to 180, the rate of turn causes the headwind component to reduce. This should make the airspeed reduce but the effect of the wind gradient reduces this loss. Ground-speed increases due to aerodynamic forces, Rk is positive and therefore airspeed actually increases. This is because the drift angle is relatively large and produces a large tangential force making the ground speed increase. A gain of airspeed in the second and third quarters is followed by a reduction in the fourth quarter but ending with a slight gain overall.
In the relatively high-G, RC dynamic soaring, Rk is dominant in the leeward turn giving an increase of airspeed whereas in relatively low-G albatross dynamic soaring, the effect of Rh is dominant giving the airspeed gain in the windward turn. Airspeed and ground-speed are affected by three factors: the tangential forces, both aerodynamic and gravitational and the rate of change of the headwind component.
11. Drift
Figure 11
The drift angle is greatest in the middle of the leeward turn at the greatest height and the greatest wind. The effect of the drift angle is to enable the centripetal forces, which are making the aircraft turn, to also act to make the ground-speed increase and thus increase component K. The drift-angle is zero at the 360 and 180 degree positions and changes from left to right drift at these positions.
Drift is also approximately zero at the starting point on the 270 degree crosswind wind-angle, at the lowest point of the circle, at the bottom of the wind-gradient where the wind is at a minimum. This will minimise the effect of Rk which is making the ground-speed reduce at this point.
Drift is either on the same side as the angle of bank during the leeward turn (positive or right in this case of a right turn) or on the opposite side during the windward turn (negative or left drift whilst still turning right in this case).
12. Rate of change of airspeed
Figure 12
All of these effects add and subtract from the rate of change of airspeed and ground-speed throughout the circle, which means that the only way to see what is going on is to plot the various changes during a full 360 circle and analyse the graphical results as in these diagrams.
The sum of these three effects gives the rate of change of airspeed Rv as shown in the figure 12 and that, in turn, leads to the actual airspeed as shown earlier. When Rv is greater than zero, airspeed is increasing and when Rv is less than zero, airspeed is decreasing.
The important point to note is that Rv is positive and the airspeed is increasing, throughout the leeward turn, the middle part of the diagram. The greatest rate of change of airspeed is at the apex of the turn on the crosswind heading. There are not two pulses of speed increase when climbing upwind or descending downwind as predicted by the Rayleigh cycle.
13. Headwind component
Figure 13
The diagram of the headwind component H (figure 13) shows how the headwind component increases during the first and last quarters, the two parts of the windward turn, (Rh or the slope of H is positive) and reduces (Rh is negative) in the middle half, the leeward turn. The last quarter is a reducing tailwind, which has the same effect on airspeed as an increasing headwind. In the last quarter, as the aircraft is descending through the wind gradient, the headwind component increases, even as the actual wind is reducing, because this part is a windward turn.
The middle section is the leeward turn during which the aircraft climbs and descends in the wind-gradient. The actual wind increases and then reduces but the headwind component only reduces. This should make the airspeed reduce but Rk, the rate of change of component K, is slightly greater than Rh and the airspeed actually increases in the leeward turn.
14. Interpretation
These results are difficult to interpret. The changing headwind component depends on both the change of height of the glider in the wind-gradient and on the change of direction relative to the wind direction. The biggest gain of airspeed is in the leeward turn where the headwind component is reducing (Rh is negative) but the effect of Rk is greatest (Rk is positive). But remember that +Rk only has to be slightly greater than -Rh to give a gain of airspeed.
I interpret this to indicate that the principle effect on the airspeed of the glider is the acceleration due to the aerodynamic forces and the angle of drift. The airspeed does not increase simply because the wind increases with height. The effect of the wind-gradient is to increase the angle of drift in the leeward turn at the top of the climb. The effect of the high-G leeward turn and the large drift angle is to allow the aerodynamic force to rapidly increase the ground-speed and thereby increase component K. The acceleration of airspeed Rv is then the sum of the rate of change of component K (+Rk) and the rate of change of component H (-Rh). Therefore Rv is positive because Rk is positive and greater in magnitude than Rh which is negative.
Therefore, airspeed increases in the leeward turn and this is what is actually seen in RC dynamic soaring.
15. The difference between RC Dynamic soaring and albatross dynamic soaring
The maths and geometry in this RC dynamic soaring illustration is essentially the same as that used to describe the flight of the albatross according to the Windward Turn Theory. The difference is that RC dynamic soaring uses a stationary hill and high-G turns in a wind-gradient to achieve high airspeeds while maintaining the position of the circle relative to the ground.
This is quite different to classic albatross dynamic soaring in which the albatrosses do not fly circles but rather fly short segments of the windward and leeward turns, at low-G and reversing the direction of the turn between each segment. The objective of the albatross is to achieve distance with minimum effort but at the cost of drifting downwind. The albatrosses leeward turn is flown as a low-G wing-over at a large angle of bank but a relatively small angle of attack. The combination of the aerodynamic forces and the angle of drift cause the ground-speed to increase throughout the leeward turn but the air-speed still reduces slightly. The amplitude of the albatross leeward turn is relatively small, only about 60 degrees before it reverses direction into another windward turn. This works quite well in a uniform wind and does not require a wind gradient, although a wind gradient does make a difference to the outcome.
16. Upwind dynamic soaring by albatrosses
However, albatrosses are thought to be able to soar upwind and the high-G leeward turn may be an explanation of how they do that, leading to a third method of dynamic soaring. If albatrosses are able to tolerate higher-G manoeuvring for a short distance, then they will be able to use this technique to make distance upwind. Which appears to be exactly what they do when homing-in on their prey. The albatross foraging strategy appears to involve detecting the scent plumes produced by their prey when close to the surface. These scent trails drifting downwind, are intersected by the long-distance, low-energy crosswind flight paths which the albatrosses use during foraging. Once the scent plume is detected the albatross must follow it upwind, either by flapping or by using the high-G dynamic soaring techniques described above but with a bank reversal at each crosswind heading. This is seen in the Laysan Albatross Tracking Data section.
17. Analogy
This is all rather more complicated than what Lord Rayleigh described in 1883. Sure, the RC glider climbs and descends through the wind gradient and gains airspeed but the process of turning adds another layer of complexity.
The easiest way to picture what is going on is to imagine you have a rock tied to a string and you are whirling it around your head, like a sling-shot. The tension in the string is like the horizontal resultant aerodynamic force FR acting on the glider in the leeward turn. Your hand moves in a circle while pulling on the string, which means there is an angle between the string and the radius from the rock to the centre of the circle described by your hand, which is analogous to the angle of drift. This allows the string to apply both a centripetal force FGC giving the rock a curved path, as well as a tangential force FGT making the tangential speed of the rock increase. Which of course increases both actual speed and airspeed.
In RC dynamic soaring the effective centre of the circle is not in a fixed position but moves around in a circular motion, which is seen by comparing the ground and air plots in which the crosswind motion (as well as the up/downwind motion) is different in the two diagrams.
It is now obvious that the basic mechanisms of gliding flight must be the same for all gliders, whatever manoeuvres they perform.
The fact that the Windward Turn Theory can be used to simulate three real forms of dynamic soaring shows the basic validity of the theory.
