Windgradient Theory
On this page I will explain why the current theories of dynamic soaring are incorrect or insufficient. The main reason why dynamic soaring has been misunderstood for so long is that the extant theories have not been critically examined and that there is simply more to dynamic soaring than just the windgradient. The biggest mistake has been the assumption that the kinetic energy of an aircraft in flight depends on it’s airspeed.
What is wrong with the Rayleigh cycle? Figure 20
See figure 20 The ‘classic’ theory of dynamic soaring is called the ‘Rayleigh Cycle’ and it works like this; The glider descends downwind and passes through a horizontal shear boundary into a layer of slower or stationary air. Actual speed is constant and airspeed increases by the same amount as the change of wind speed. The glider then turns onto an upwind heading and climbs back up through the shear boundary and the airspeed again increases by the same amount as the change of the wind speed. The glider then turns downwind and repeats the process. If the gain of airspeed in the wind gradient is equal to the losses due to drag in the turns, then airspeed converts to height and height can be maintained or gained. There are many things wrong with this:
1: The theory is impractical
If a bird, maintaining constant actualvelocity, passes from one layer of air to another and each layer is moving at constant velocity, then there is no acceleration and no gain of energy. The only change is a different value of airspeed and drag in each layer. When descending downwind through a wind shear, actual velocity will only be constant if the rate of descent increases to overcome the extra drag.
When climbing upwind through a wind shear, against gravity and an increasing headwind, actual velocity cannot be constant and the gain of airspeed must be less than the difference of velocity between the layers.
2: The theory does not account for the depth of the shear boundary or for drag when passing through the shear boundary
The shear boundary must have a finite depth and in climbing and descending through it there must be an exchange of potential and kinetic energy. In climbing through the shear boundary, there is a gain of potential energy and airspeed must reduce before there can be a gain of airspeed from the shear wind. Actual speed then increases when the downwind turn is made. When downwind and descending through the shear boundary, the same amount of potential energy as was gained in the climb is now converted to kinetic energy. But because the actual speed is now greater and KE is a square law, the given kinetic energy is only worth a smaller speed increment. Therefore, overall, actual speed is lost.
3: The theory is not a good model of the atmosphere or of bird flight
To achieve height gain the thin shear boundary would have to be repeated continuously as the bird climbs which is not a good model of actual atmospheric wind gradients. Lord Rayleigh knew this and said as much in his 1883 paper;
‘A priori I should not have supposed that the variation of velocity with height to be adequate for the purpose; but if the facts are correct [Peel’s observation], some explanation is badly wanted.
The theory is not a good model of albatross dynamic soaring in which the windward and leeward turns are of a distinctly different shape and the amount of turn is nowhere near 180 degrees.
4: The wind gradient theory as described by Rayleigh and developed by Lissaman assumes that between moving air masses there are improbably thin shear layers and therefore improbably steep wind gradients. It assumes the bird can climb and descend through a shear boundary with no gain or loss of height, no exchange of PE and KE, with no turbulence and no drag. Although airspeed and drag may increase there is no increase in total kinetic energy and no extra force in the direction of flight to overcome the extra drag.
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How much energy is in the wind gradient?
What is the minimum windgradient needed to overcome gravity?
Suppose the wind gradient is a change of wind speed of w in a height band of h .The wind gradient is W_{g}=w/h s^{1}
If the bird starts at airspeed v_{0}_{ }and height h_{ }dives through the wind gradient and increases to airspeed v_{1}=v_{0}+ w in height of h, that excess speed converts to height according to the energy exchange given by mgh= ½ mv_{1}^{2} ½mv_{0}^{2}
therefore h= (v_{1}^{2} v_{0}^{2})/2g
For a gain of height of 1m, h=1 and say v_{0}=10m/s the minimum necessary gain of speed will be
v_{1}^{2}=v_{0}^{2}+2g
v_{1}^{2}=10^{2}+2x9.81
v_{1} =10.94ms^{1}
w= v_{1}v_{0}
w=0.94 ms^{1}^{ }
This means that if the gain of height is to be equal to the height lost gaining speed then the wind gradient would need to be 0.94ms^{1} per metre of height. That is 3.4km/hr per m of height.
What are theoretical windgradients like?
Some authors advocate the windgradient theory using a twolayer windgradient model in which the lower layer is stationary. They then create a mathematical model in which the the wind gradient is the same as the wind and the bird gains airspeed equal to the wind with each passage through the shear boundary. The bird then loses this airspeed during each turn.
This is completely unrealistic. The useable wind gradient can only ever be a proportion of the whole wind. Actual speed cannot be constant and the bird cannot gain the full amount of the change of the wind in the wind gradient. The lower layer would need to be deep enough to accommodate a banked turn which could only occur occasionally in the lee of a breaking wave and not on a continuous basis. Any such calms in the troughs between swells will be moving with the swells and most probably downwind making upwind travel very difficult.
What are real windgradients like?
Actual average wind gradients are reported to be more like 0.25 ms^{1} per m of height but it depends on which model of wind gradient you use. In aviation terms the wind gradient occurs below about 700m, the wind approximately halves from 700m down to 10m (the height at which the wind is measured at an aerodrome). The wind at 10m is considered to be representative for an aircraft taking off or landing, say down to a wing height of 1m. Watching smoke blowing over grass you will see a marked boundary layer below about half a metre. When landing an aircraft the wind gradient is most noticeable below about 100m. These effects will be less over water than over land due to less surface friction. ‘You pays your money and you takes your choice’.
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How much height could be gained from a typical windgradient?
Looking at it another way, how much height could be gained from a typical wind gradient? Consider a bird gliding at 10ms^{1} in still air. At constant speed and rate of descent the bird is ‘propelled’ by gravity and has constant kinetic energy (KE) just like a vehicle freewheeling down a hill. All of the potential energy (PE) in its height is used up overcoming drag. At the end of the glide, there is no excess airspeed available for a zoom climb. Now consider the same bird gliding with a tailwind from a height of 21m to a height of 1m. Assume the wind at 21m above the surface is 15ms^{1 }and at 1m the wind is 10ms^{1 }.The wind gradient would then be 5ms^{1} in 20m i.e. 0.25ms^{1} per m of height. If the bird is able to increase airspeed by 5 ms^{1 }by transfer of momentum from the wind gradient then a zoom climb would be possible. In order to gain the airspeed from the wind gradient, the ground speed in the dive is constant.
The energy transfer from speed to height is given by
KE=PE
½ mv_{1}^{2} ½mv_{0}^{2} = mgh m is the mass of the bird v_{0} is the bird’s initial airspeed v_{1 }is the bird’s final airspeed having gained 5 m/s in the wind gradient h is the height gain
½(15^{2}10^{2})=9.81h
h=6.37m
This would be a gain of height of 6.37m but minus drag losses consequent to the increased airspeed and any turning forces. If the bird continued downwind it would climb into an increasing tailwind and lose the speed gained in the dive. If it turned across the wind it would gain just the 6.37m. If it turned into a headwind it would potentially gain another 6.37m but again less drag losses due to the turn. Anyway you look at it, the bird ends up short of the 21m height it started with. Therefore dynamic soaring cannot be only about the wind gradient. There must be a different mechanism involved.
A free lunch
Of course, on the other hand, if the bird encounters that 5m/s as a random horizontal gust it could potentially gain the same 6.37m of height as a free lunch, but only if the energy transfer is achieved by direct transfer of momentum without an actual increase of airspeed and drag. If the bird reacts fast enough, it experiences the gust as a propulsive thrust.
Then again, negative gusts would have the reverse effect….
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What is wrong with the Lissaman Loop?
The Lissaman Loop is an attempt to account for the drag losses not accounted for in the Rayleigh cycle and is a development of that hypothesis. It was published by the late Dr Peter Lissaman in April 2007 in a paper called Fundamentals of Energy Extraction From Natural Winds.
Figure 21
The proposition is that energy can be extracted from the wind in a downwind turn and in descending downwind and climbing upwind through a wind gradient. It works like this, see figure 21:
The downwind turn is like flying around the inside of an alcove moving downwind at say 2 units of speed.
The bird begins with groundspeed 8 units, flying upwind with headwind 2 and therefore with airspeed 10.
The pinball albatross enters the alcove, loses 2 units due to drag and emerges from the alcove flying downwind at airspeed 8, tailwind 2 and groundspeed 10.
The bird now descends through a thin shear boundary into the stillair layer below, maintaining ground speed 10 and airspeed becomes 10, gaining 2.
The bird now flies around the inside of the ‘windward’ alcove in still air, loses another 2 units due to drag and emerges with groundspeed and airspeed 8.
Now the bird climbs back through the shear boundary gaining 2 units from the wind so that groundspeed is 8 and airspeed is 10 again.
What is wrong with that?
1: Well, this is supposed to be energy neutral, that is it begins and ends with the same airspeed and height although there is a loss of distance downwind during the downwind turn. However it assumes an improbably thin shear boundary and therefore in improbably steep wind gradient with no change of height, no change of potential energy, no turbulence and no losses due to drag in the shear boundary.
In practice, descending, whether through a shear boundary or not, will always result in a loss of height and either a gain of speed or if no speed is gained then a dissipation of energy due to drag.
When descending through the shear layer, 2 units are gained due to the wind shear but some energy must be lost due to drag. Groundspeed must then be less than 10 and, at the end of the turn, less than 8. In climbing up through the shear boundary there will again be some loss due to drag and due to the gain of PE during the climb, before gaining 2 units due to the wind gradient, so airspeed is less than 10.
Furthermore as the descent is made at groundspeed 10 and the climb at ground speed 8 there is a further loss of distance downwind. This is nowhere near energy neutral.
2: The Lissaman loop is supposed to be a model of dynamic soaring but it is clearly not that. In the Lissaman Loop the two turns are essentially the same, but albatross clearly make their downwind and windward turns of different shapes and at different speeds and with much less than 180 degree turns.
3: To make the model work Lissaman says that extreme values of G loading (5G+) and bank angles of 70 degrees are required but albatross are known to fly with low metabolic rates implying minimum effort. Also it can be shown that while 70 degree banked level turns as required by the Lissaman Loop require 3G, a steep ‘wingover’ can be flown with little more than 1G
The Downwind Turn
If a glider is heading into wind it has low groundspeed and low kinetic energy. As it turns downwind groundspeed and kinetic energy increase. Where does the energy come from? The glider is propelled by gravity and the energy comes from an increased expenditure of potential energy, that is to say height. In other words an increased rate of descent. It is not propelled by the wind. The force between the glider and the air is drag opposite to the direction of flight (and lift which balances the weight and the turning forces). In a normal turn with a moderate angle of bank and a small angle of drift the aerodynamic forces are not strong enough to be propulsive. In turning downwind, the glider is not propelled by the wind, at constant airspeed it is diving to gain groundspeed to keep up with the wind.
Why is it thought that dynamic soaring is about the use of the wind gradient?
Dynamic soaring was first considered by Lord Rayleigh in 1883 and he asserted in an article in Nature that birds cannot soar in a uniform, horizontal wind. People have been trying to explain dynamic soaring in terms of the wind gradient ever since. In 1883 mankind had not yet learned to fly and had not discovered that the effect of wind gradients can be counterintuitive.
Perhaps this misunderstanding about the role of the wind gradient has come about because the windgradient can improve the efficiency of the leeward turn without the windgradient being the actual source of energy. The ground effect can improve the efficiency of the windward turn and combined with the wind gradient and the bird’s desire to achieve distance rather than height, is the reason the bird stays at lowlevel.
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Gust Soaring
An alternative theory is called gust soaring . See Figure 22. A glider’s rate of descent is related to it’s airspeed by the lift/drag ratio which varies with angle of attack and airspeed. The sink rate and airspeed can be plotted on a polar diagram. The best angle of glide is the tangent from the origin, whilst the rate of descent is at a minimum at a particular airspeed and angle of attack, (the minimum sink airspeed). The polar diagram relates sink rate and airspeed at a load factor of one (1G). Each airspeed has a corresponding angle of attack so that lift equals weight and drag increases with the square of the airspeed. The polar diagram tells you the sink rate corresponding to each airspeed at 1G
Figure 22.
The gust soaring theory says that when a bird encounters a vertical gust, it is the same as an increase in the bird’s sink rate, which results in an increase in airspeed according to the polar diagram, which excess airspeed can then be converted to height. According to this theory a bird gliding at say 15m/s (sink rate 0.68m/s) encountering a vertical gust of say 1m/s would then have a sink rate of 1.68m/s and would accelerate to the corresponding speed on the polar diagram of say 25.5 m/s and convert that excess speed to height gaining 21.7m. (Pennycuick 2002)
This is incorrect because the effect on the bird of the ventral gust is to increase the angle of attack, which effectively increases the load factor by increasing the coefficient of lift. The polar diagram is no longer valid and there is no reason why the airspeed will automatically increase to any particular value. An increase of airspeed as in the example above would require a sustained force in the direction of flight. Apart from the pulse (below), the only force available is gravity and that will require a loss of height.
However there is a mechanism that explains the gain in airspeed, the surge that the glider pilot experiences when he flies into a vertical gust like a thermal. What happens is that the increased angle of attack causes an increase in the lift and drag components respectively normal to and in line with the new relative wind direction, while the glider continues moving with its original velocity. The new resultant force is now tilted forward relative to the glider’s actual flight direction and can be resolved into two new forces, thrust in the direction of flight which causes the airspeed to increase, and a force at right angles which is like lift but is actually a centripetal force which causes the glider to pitch up in a curve. The vertical motion of the glider diminishes the effect of the gust and equilibrium is restored. The glider simply flies out of the gust having gained a little speed due to the pulse of acceleration. This resolution of the forces on the wing is similar to how a helicopter rotor works during auto rotation
See Figure 23 The real effect of a vertical gust is to increase the angle of attack and the airspeed by a small amount, giving a small pulse of energy
The increased drag pulls the airspeed back and equilibrium is restored. At an airspeed of 15m/s, a 1m/s vertical gust will cause a small increase of angle of attack and an increase of airspeed of about 0.08m/s but only briefly.
On the other hand, a sustained vertical component of wind velocity added vectorially to the gliders sink rate will give a sustained modification of its flight path.
If the vertical air current is sustained then the glider will achieve an identical rate of climb (minus its sink rate). A vertical current of 1m/s will give a net rate of climb of 10.68=0.32m/s which, if sustained for 67.8 s would then gain the bird 21.7m of height.
Alternatively if the bird wants to maintain height in such a 1m/s updraught, it can dive to increase its airspeed such thatits sink rate is 1m/s down at 1G according to the polar diagram, which will then equal the vertical wind current.
If the vertical current is caused by a passing ocean wave or a thermal then this would be a valid method of soaring similar to hill soaring practiced by birds and glider pilots over land. Such an effect can be seen when pelicans soar at constant height, along a swell approaching a shore with perhaps an offshore breeze.
Similarly if the bird is in a vertical banked turn and briefly penetrates a shear boundary giving it a horizontal gust (ventral from the birds point of view) it will get a pulse of thrust giving it an incremental increase of airspeed. It will not be an increase of airspeed derived from the polar diagram.
Conclusions
The Rayleigh/Lissaman cycle of dynamic soaring is incorrect because
1: It is impractical
2: It does not account for drag when passing through the shear boundary
3: It does not account for the depth of the shear boundary
4: It is not a good model of bird flight or of the atmosphere
5 It assumes that between moving air masses there are improbably thin shear layers and therefore improbably steep wind gradients, with no turbulence and no exchange of potential energy.
6: It is not energy neutral.
7: In turning downwind, a glider is not propelled by the wind, it is diving to gain groundspeed to keep up with the wind.
8: Gust soaring is a misapplication of the glide polar curve.
9: Lee soaring is better explained by the aerodynamics of autorotation.
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Click here for the next page which is a critique of Lord Rayleigh’s 1883 article in Nature
