The Windward Turn Theory of Dynamic Soaring

 This page contains more detail about the windward and leeward turns and the transfer of momentum and energy.

The Dynamic Soaring Manoeuvre

 The windward turn starts with a tailwind component at point A (figure 1) and ends with a headwind component at point B. There is a reversal of the direction of turn and the leeward turn starts with a headwind at point B and ends with a tailwind at point C. The direction of turn reverses again before the next windward turn. Figure 1 is not to scale but shows the typical shape of the manoeuvre. The windward and leeward turns can be of a similar length or the windward turn could be three or four times longer than the leeward turn depending on the wind conditions. The overall length of the manoeuvre is of the order of 200 to 400 metres. The lowest point of the windward turn is skimming the surface while the highest point of the leeward turn is 10 to 20 metres. The average heading is approximately crosswind and varying by +/- 20 to 30degrees. The amplitudes of the windward and leeward turns are the same in order to maintain a constant average heading relative to the wind. The effects which enable dynamic soaring are at a maximum close to the crosswind heading and diminish if the heading gets too far upwind or downwind. Because the heading is approximately crosswind, the drift angle is at a maximum, therefore the average track has a large downwind component. Click here for an animated view of the dynamic soaring manoeuvre.

Figure 1

 

The Windward Turn

 In the windward turn, the albatross maintains height and loses momentum due to the unbalanced horizontal aerodynamic force. The loss of momentum is seen as a loss of ground-speed rather than a loss of airspeed. Airspeed is constant, or increases slightly, because the tendency to lose airspeed due to drag is balanced by the tendency to gain airspeed from the increasing headwind component, whilst turning relative to the wind. An increasing headwind component has the same effect on the airspeed as a decreasing tailwind component.

In the windward turn, how do the velocities change?

 Figure 2 is a plan view of the windward turn showing the triangles of velocity at four points. Note the drift angle d and the wind-angle y. It can be seen that the wind-velocity is uniform and that the wind-speed is a large proportion of the birds airspeed. The amount of turn is limited to less than about 60 degrees. During the windward turn there is a large change of wind-angle and of headwind-component from tailwind to headwind. The triangle of velocities itself does not cause the velocities to change, it is simply a way of depicting the three velocities at one moment in time. Those velocity changes are caused by applied forces like lift, drag or gravity.

 The bird maintains height and loses ground-speed and momentum due to aerodynamic forces. The loss of momentum is seen as a loss of ground-speed but not a loss of airspeed. The wind-speed and direction are constant but the direction of the bird relative to the wind, the wind-angle, is changing. Note how the shapes of the triangles are continuously changing due to the bird turning relative to the wind and that the ground-velocity (actual velocity) is reducing as the wind angle y reduces. Note also that the drift angle d is almost constant in the range of wind-angles used by the albatross

Figure 2

Maintaining Airspeed

 How can the bird, in level flight, maintain its airspeed? Quite simply, you can see in figure 2 that, as the wind-angle y reduces, the tailwind component reduces and then the headwind component increases causing a tendency for the airspeed to increase. At the same time, due to aerodynamic forces, the ground-speed reduces causing a tendency for the airspeed to decrease. When these two effects are equal and opposite, the airspeed is constant. In other words, if the ground-speed reduces due to aerodynamic forces at the same time as the wind angle reduces due to the birds rate of turn, then there is no reason why the airspeed should change. This requires a little more explanation.

Equilibrium

 We know that, in a state of equilibrium, when all velocities are constant, air-velocity is maintained by a balance of thrust and drag (for a glider, a balance of a component of weight and drag). Ground-velocity is the vector sum of air-velocity and wind-velocity.

 When the wind is changing from one state of equilibrium to another, the air is accelerated and the aircraft air-velocity will change by a small amount (as in turbulence) causing changes to the aerodynamic forces. The changing forces then cause the ground-velocity to change, until a new state of equilibrium is achieved and the aircraft  has returned to its original airspeed with the drag balanced by thrust or for a glider, a component of weight. For example, if the headwind increases, airspeed increases slightly while ground-speed is initially unchanged due to inertia resisting any change of ground-velocity. As drag increases, ground-speed and airspeed reduce together and a new state of equilibrium is reached with the original airspeed but with reduced ground-speed corresponding to the new headwind component.

Equivalence

 In aerodynamics it does not matter whether the air is stationary and the aircraft moving, as in free flight, or whether the aircraft is stationary and the air is moving, as in a wind tunnel.

Turbulence

 In turbulence, sudden, brief acceleration of the air results in corresponding changes in air-velocity while changes of ground-velocity (acceleration of the aircraft) are resisted by inertia.

Acceleration

 Airspeed is relative speed and is affected by both acceleration of the wind and acceleration of the aircraft. Either acceleration can be either centripetal or tangential, so that changing headwind-components can be due to changing wind-velocity or changing aircraft-velocity relative to the wind, eg. changing wind-angle or changing heading.

 Consider for example, a winch-launched glider accelerated during launch into a headwind. Imagine at the same time the wind itself is increasing causing an increasing headwind. The airspeed will be the sum of the ground-speed and the headwind component. The acceleration of airspeed will be the sum of the acceleration of the ground-speed and the acceleration of the headwind.

Effect of the wind during acceleration

 If thrust and drag are not equal then there will be an unbalanced force acting on the aircraft which will cause the aircraft to accelerate and both the air-velocity and ground-velocity will change. When there is no wind, the air-velocity and ground-velocity are the same and both change at the same rate. However when there is a wind, there is an angle of drift and the ground-velocity is different from the air-velocity.

 A simple example of this is as follows See Figure 3: When a glider maintains height on a constant heading, the unbalanced drag-force F₁ causes the air-speed to reduce. The force and the velocity vector are in-line and therefore the magnitude of the velocity vector changes but not the direction. If the wind is constant, as the airspeed reduces, the angle of drift will increase and the ground-velocity vector will change in both speed and direction. This is because the force and the ground-velocity vector are not in line. The change of ground-speed and direction can be calculated using components of the unbalanced drag force. The glider is in straight and level flight relative to the air but as the airspeed reduces, the drift angle increases and the path over the ground turns to the right. The wings are level and it is not turning relative to the air but its path over the ground is a curve. That curve is caused by a centripetal force component F_c.

Figure 3

 Force F₁ is the drag force causing acceleration of the airspeed. Force components F_t and F_c correspond to tangential and centripetal accelerations of the ground-velocity. F_t is less than F₁ and therefore, in this particular case, the rate of ground-acceleration is less than the rate of air-acceleration. This makes sense because, although the airspeed can in theory reduce to zero, the ground-speed can only reduce to the value of the wind-speed.

 This means that a uniform wind causes different tangential and centripetal components of acceleration relative to air-velocity and ground-velocity (air and ground frames of reference).

 It should be noted that if the air and ground frames of reference are treated as inertial frames of reference, that is that they are moving relative to each other with uniform velocity, then the TOTAL force and the TOTAL acceleration are the same in both frames (because the acceleration of the wind-velocity is zero). However, the tangential components in each frame are different from each other and the centripetal components in each frame are also different from each other.

 These effects are caused by forces acting on the glider and it is the glider which is accelerating, not the ground. Therefore while ground-speed itself has no effect on the glider, the forces acting on the glider do affect  the glider and the effect is seen as acceleration of ground-velocity

 (Force F₁ reduces as the airspeed reduces but for simplicity here can be regarded as constant. Also, this particular case is not dynamic soaring which involves both centripetal and tangential acceleration of the air-velocity)

Force components

 In dynamic soaring the bird is never in a state of equilibrium, it is always accelerating and of course, there is no thrust. In figure 4a, which shows two successive points, the bird is turning relative to the wind and maintaining height in a dynamic soaring windward turn. In a banked turn, the the lift force has a horizontal component which, combined with the drag force, effectively changes both the magnitude and direction of the horizontal acceleration. Force F₁ is drag and force F₂ is the centripetal force, the horizontal component of lift due to the angle of bank. The vertical component of the lift force, equal to weight,  is not shown.

 F₁ and F₂ are combined to create horizontal resultant force F₃ which maintains its orientation relative to the bird as it turns.

 At Position 2, an increment of time later, there is a different solution to the triangle of velocities. The heading has changed and the wind-angle and ground speed have reduced but the airspeed and the wind-velocity are the same as at Position 1.

 Figure 4a

Airspeed and ground-speed components

 See figure 4b. Force F₃ can now be divided into new components: tangential force F_t and centripetal force F_c, oriented to the direction of the ground acceleration. The ground-speed reduces due to the tangential component force F_t and the ground direction changes due to force F_c.

 Under acceleration, the airspeed V comprises, in effect, a ground speed component K and a wind speed component H. The acceleration of airspeed is the sum of the rate of increase R_H of headwind component H and the rate of decrease R_K of ground-velocity component K._.The rate of change of airspeed is zero because the rate of increase of the headwind component equals the rate of reduction of the ground-speed component.

 R_H - R_K = 0

 In other words it is not air-speed V which is reducing due to force F₁, it is the component-of-ground-velocity K which is reducing because ground speed G is reducing due to force Ft. (This point becomes more important when explaining the increase of ground-speed in the leeward turn)

 Figure 4b

 Comparing Figures 3 and 4b, it can be clearly seen that, in figure 3 in straight flight when wind-angle y is constant, that headwind component H is constant and is therefore unaffected by force F₁. Logically there is no reason why force F₁ should affect component H even during a turn when wind-angle y is changing. Therefore force F₁ only affects component K,  both in straight flight and in a turn when the increase of headwind component is due the change of wind-angle. For every wind-angle there will be an angle of bank and a rate of turn that gives a rate of increase of headwind component equal to the rate of reduction of ground-speed-component and keeps the airspeed constant.

 Imagining constant airspeed is the easiest way to visualise what is going on here. However, in reality R_H and R_K are never exactly equal and opposite and there is always a slight increase of airspeed in the windward turn and a slight decrease in the leeward turn.

 Note also that, in this context, a reducing tailwind component has the same effect as an increasing headwind component, so that the desired effect is achieved throughout the windward turn from tailwind to headwind.

Controlling height and airspeed with rate of turn

 The albatross probably judges height above the surface visually and senses airspeed by dynamic pressure through tube-nostrils. However the sea surface is constantly rising and falling and therefore the bird must climb and descend to maintain constant height above the surface to take advantage of ground-effect. This changing true height will result in changing airspeed and to maintain airspeed the bird can alter its rate of turn by changing its angle of bank.

 If airspeed reduces due to the bird gaining height, the bird increases its rate of turn toward the wind by increasing its angle of bank. If airspeed increases, as for example when descending into a trough, it reduces its angle of bank and reduces its rate of turn.

 As the windward turn proceeds, the wind angle reduces, the rate of change of the headwind component inevitably reduces and ultimately the albatross will not be able to maintain its airspeed. Before this happens (Figure 2 point P4), the albatross can increase its rate of turn to gain a margin of airspeed and height before reversing the direction of turn into the leeward turn. The penalty here is a small increase in drag, which reduces the total distance flown in the windward turn.

Drag losses are balanced by loss of kinetic energy equivalent to potential energy

 In still air and at constant velocity and drag, a glider loses height according to its glide ratio ( the same as its lift/drag ratio). An albatross has a glide ratio of about 1:20, meaning it loses one metre of height for every 20 meters flown. Therefore all drag losses, at constant airspeed, can be expressed as an equivalent loss of height or potential energy.

 Airspeed is approximately constant in the windward turn, but kinetic energy at the beginning of the turn is proportional to airspeed plus tailwind component (squared) and at the end of the turn is proportional to airspeed minus headwind component (squared). The law of conservation of energy means that energy is not lost but can be converted to another form of energy. The change of kinetic energy (KE) is equivalent to a change of potential energy (PE) which, in turn, is equivalent to drag losses. When the wind is strong enough, the drift angle allows the aerodynamic force to increase the KE and PE more than the losses due to drag and the bird can maintain or gain height.

 Although velocity and KE are frame dependent, dynamic soaring depends upon rate of change of momentum ie acceleration, which is not frame dependent.

 During the windward turn the bird maintains approximately its best glide speed. A small increase in airspeed will increase stall margins and ensure that there is sufficient excess height or airspeed to complete the leeward turn.

Why does the bird reverse the direction of turn?

  At the end of the windward turn the bird has maintained airspeed at the cost of losing ground-speed. Maintaining airspeed depends upon the ability to match the rate of loss of ground-speed component with the rate of increase of headwind- component, which depends upon the rate of change of headwind-component relative to wind-angle. The crosswind position is where the rate of change of headwind with respect to the wind-angle, is at a maximum.  As the turn proceeds and the wind-angle reduces, the rate of change of headwind component reduces, whilst drag and the rate of change of ground-speed is approximately constant (Figure 2 points P3 & P4). The end of the windward turn comes when the birds ability to maintain airspeed and height diminishes. It must now regain ground-speed and it can only do this by turning downwind.

 At the end of the leeward turn the bird has restored its ground-speed and momentum but has run out of height and airspeed. It must again reverse its direction of turn and start another windward turn.

Limits on the energy gained in each turn

 The amount of KE expended in the wind-ward turn and hence the distance flown, depends on the airspeed and the tail/headwind component at each end of the turn. This implies that maximum energy gain would mean 180 degree turns. However, the ability to maintain height depends on the rate of change of head/tailwind component, which is greatest in the middle part of the windward turn and diminishes when turning to within about 50 degrees of the wind. So there are severe limits on how close to the wind the bird can fly and how much energy the bird can extract from the wind.

Ground Effect

 During the windward turn the bird flies close to the surface to gain advantage of the ground-effect. This is where the lift-induced drag of an aircraft is reduced by the close proximity of the surface.

 The lift and drag forces are the equal and opposite reactions to the downward and forward momentum given to the air by the wings down-wash. The vertical momentum given to the air is always equal to the weight in straight flight. However, when the wing is within about half a wingspan of the surface, the forward momentum given to the air is reduced and it produces the same lift with less of a drag penalty. This does not add energy but improves the efficiency of flight in ground-effect, increases the distance flown and provides the incentive to the bird to remain close to the surface.

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The Leeward Turn

 The albatross gains momentum in the leeward turn using a component of aerodynamic force to act as a propulsive force in the direction of the ground-velocity. This component provides the acceleration which is seen as an increase in ground-speed rather than airspeed. Thus it gains horizontal momentum and kinetic energy without losing potential energy other than a small drag loss during the turn reversals. This propulsive force is a component of the horizontal resultant which, in turn, is the vector sum of the horizontal, centripetal component of lift and the drag force.

 Figure 5 is a plan view of the leeward turn with the wind coming from the top. Notice how the shape of the triangle of velocities changes with the changing wind-angle and acceleration of ground velocity. It can be seen that when the wind is a large proportion of the airspeed, the drift angle d (the angle between the aircraft heading and its actual track across the ground) is also large. Force F₃ is the horizontal resultant, the vector sum of the drag force F₁ and the horizontal component of lift F₂. Because of the large drift angle, there is now a propulsive component F_t acting in the direction of the ground-velocity and it is this which increases the birds ground-speed. Component F_c causes the curved path relative to the ground. This is essentially the same mechanism as in the windward turn but with a propulsive effect on ground velocity instead of a retarding effect.

 Component F_t  rapidly diminishes as the bird reaches a wind-angle of about 130 deg and the drift-angle reduces, at which point the bird can no longer gain ground-speed without losing height and it reverses the direction of turn into the next windward turn.  When the amount of turn, the amplitude of the turn, is the same in both the windward and leeward turns, then the average ground track is constant.

Figure 5

 In dynamic soaring, the leeward turn is a wing-over or arched turn, which enables a large angle of bank to provide a large horizontal component of lift, without actually increasing the total lift force and without increasing the drag loading. The airspeed is shown as constant but in practice, as this is a wing-over and height is gained and lost, there will be a corresponding loss and gain of airspeed.  When the bird is gaining height, the propulsive force F_t has a vertical component and the bird will gain extra height and potential energy. The wing-over gives the bird time to accelerate, to keep-up with the wind as it turns downwind.

How is airspeed the same at the beginning and end of the leeward turn?

 As in the windward turn, in the leeward turn the airspeed V effectively comprises a ground-speed component K and a head/tail-wind component H. When the acceleration of the ground-speed component K is equal and opposite to the acceleration of the wind component H, the acceleration of airspeed is zero. see figure 6

Figure 6

 In this case, the ground-speed G is increasing under the effect of the propulsive force F_t and component K increases at the same time. Meanwhile the headwind component H reduces under the effect of the increasing wind-angle y.

 It is difficult to understand how airspeed can be constant when there is an unbalanced drag force; we are used to the idea that drag causes loss of airspeed. However, in this case the drag force is part of the horizontal resultant F₃ which causes acceleration of the ground velocity (speed and direction) which then causes acceleration of component K which is part of the air-velocity V

 In practice, the rates of change of components H and K are never exactly the same in the range of wind-angles used by the albatrosses. The magnitude of R_H is always slightly greater than that of R_K. Therefore the manoeuvre will lose a small amount of airspeed in the leeward turn and increase ground-speed to make up the loss in the windward turn. This is due to the different shapes of the windward and leeward turns and the different rates of exchange of momentum.

Wind Gradient

 During the leeward wing-over there is an exchange of airspeed to height and back to airspeed again, while ground-speed is being gained. However, if there is a wind-gradient, this will cause the headwind component to increase slightly both during the climb upwind and the descent downwind which will offset some of the net loss of airspeed due to drag but it will not necessarily increase the airspeed. It will only be a small effect because the bird is never near to an exactly upwind or down wind heading.

Airspeed Sense

 The need to maintain or slightly increase airspeed in the windward turn and the need to sense when airspeed is changing, is critical and controls the birds management of the manoeuvre. The points at which the turn-reversal from the windward to leeward turn and vice-versa depend upon the birds ability to sense when the airspeed (dynamic pressure) and height (visual) relationship is changing.

 This may explain why birds that do dynamic soaring for a living are found to have nostrils of the tube nose sort. They may make these birds particularly sensitive to dynamic pressure and therefore to rate of change of airspeed. I suspect that, in other forms of bird flight, angle of attack and ground speed are more important than pure airspeed.

 Tube nostrils are similar to the pitot tubes found on aircraft. Pitot tubes are the forward facing vents which feed dynamic air pressure, which is proportional to airspeed, to the airspeed indicator. In practice pitot pressure is the sum of dynamic and static pressure. Static pressure varies with height and from day to day, so that pitot pressure is not exactly the same as airspeed. However, the rate of change of pitot pressure will give the bird a sense of the rate of change of airspeed. The picture shows a giant petrel and its tube nostrils.

Transfer of Energy and Momentum

 How exactly is energy exchanged between the wind and the bird? Fundamentally, the process is a transfer of momentum, the same principle as colliding pool balls. Momentum is proportional to velocity whilst kinetic energy is proportional to velocity squared. The momentum of the wind depends on wind-speed and that is measured relative to the ground. This does not make the ground a privileged frame of reference. It is simply a convenient,  common frame of reference relative to which to measure the wind-velocity and the birds velocity and hence its acceleration. The birds velocity is, of course, the vector sum of its air-velocity and the wind-velocity. Also, the wind is caused by variation of temperature and pressure within the atmosphere. When the wind gains speed and momentum, it is the air which is accelerating and not the ground.

 In flight, a wing gives the air downward and forward momentum. The equal and opposite reaction is an upward and backward force. The upward component is called lift and the horizontal component, opposite to the direction of flight, is called drag. When an aircraft banks, the lift force tilts with the aircraft and provides vertical and  horizontal components. The horizontal component is the centripetal force which makes the aircraft turn. That horizontal, centripetal component of lift added vectorially to the drag force, creates the horizontal resultant force F₃. The angle of bank therefore gives horizontal momentum to the air in the opposite direction to the horizontal resultant force. See figure 7

Figure 7

 In the leeward turn, a component of that horizontal momentum given to the air is parallel to and opposite to the wind direction; therefore the momentum of the wind is reduced, whilst the  momentum of the bird is increased. The forces are relatively large due to the large angle of bank and therefore, the rate of change of momentum is relatively quick but the length of the leeward turn is relatively short.

 During the windward turn, the opposite effect occurs and part of the horizontal momentum given to the air is in the same direction as the wind and increases the momentum of the wind. The wind gains momentum and the bird loses momentum.  The windward turn is flown with a small angle of bank and a small horizontal component of lift and therefore the turn is long and the exchange of momentum is at a slow rate.

 The momentum in both turns and for both the bird and the air is measured relative to the same ground frame of reference.

 The bird maintains average speed and height during successive turns and its mass is of course constant. Therefore for the bird, the change of momentum is the same in each turn. However the leeward turn is shorter than the windward turn and less air is given more acceleration compared with the windward turn in which more air is given less acceleration. So, while the total change of momentum is the same, each unit mass of air loses more speed and energy in the leeward turn than each unit mass of air gets back in the windward turn. This is because, in relation to velocity, momentum is a direct law and kinetic energy is a square law. The difference in energy is equivalent to the birds drag losses. In effect, wind speed has been converted into air turbulence in the wake of the bird. That turbulence energy is then dissipated as heat at the molecular level. The horizontal momentum taken from or given to the air affects only the air with which the bird is in contact and, of course, the unit masses of air encountered in each turn are different. That momentum is then dissipated through the greater mass of air through which the bird is flying so that the overall effect on the wind is extremely small.

 This explains why the windward and leeward turns are of different types. In the windward turn the rate of turn is slow to gain the maximum distance and advantage is gained from the ground-effect. In the leeward turn, the advantage of the ground-effect is given up in favour of a steep angle of bank and a quick rate of turn.

Frames of Reference

 Some people have objected to this hypothesis because velocity, momentum and kinetic energy are frame-specific. In other words these parameters can be measured relative to any Inertial Frame of Reference (IFR) and one IFR is as good as another. An IFR is a frame of reference which is moving with uniform velocity that is, not subject to acceleration or rotation.

 We are explaining flight on Planet Earth and the only practical frame of reference against which to to measure wind-velocity and bird-velocity is the ground. Therefore the ground can be treated as an approximate IFR for the puposes of these calculations. However, the fact is that the ground (the surface of planet Earth) is a rotating curved surface and any object moving across the surface is subject to gravity and is moving in a curve. This means that velocity is really angular velocity and momentum is really angular momentum.

 In that sense, ground-speed is real as opposed to any other speed measured relative to an imaginary IFR

Summary

1: Albatross dynamic soaring comprises alternating windward and leeward turns. The average heading is crosswind plus and minus about 20 to 30 degrees.

2: The windward turn is flown at a small angle of bank at approximately constant height above the surface. Ground-speed reduces due to the retarding effect of the aerodynamic forces. The albatross can maintain airspeed because the tendency to lose airspeed and momentum due to drag is balanced by the tendency to increase airspeed caused by the increasing head-wind component. These changing tail/headwind components are due to the bird turning relative to the wind.

3: The leeward turn is flown as a steeply banked wing-over which enables a large centripetal force at about 1G load-factor. Ground-speed increases due to the propulsive effect of a component of the aerodynamic force. Loss of airspeed is minimised by the approximate balance of reducing headwind component and increasing ground-speed component.

4: Ground-effect reduces drag in the windward turn while the wind-gradient reduces airspeed losses in the leeward turn. The birds foraging strategy is to gain distance rather than height. These three factors explain why albatrosses fly at low level.

5:  The birds average airspeed and height is constant, so that it does not gain energy overall. The energy exchange involves wind-speed energy being converted to turbulence due to drag in the birds wake. This is achieved through an exchange of momentum between the bird and the wind.

6: In the leeward turn, when the bird gains momentum the wind loses momentum and vice-versa in the windward turn. Provided that the bird gains at least the same momentum in the leeward turn than it loses in the windward turn then it can maintain or gain airspeed and height.

7: The wind loses more speed energy in the leeward turn than it gets back in the windward turn, the difference being the drag energy. Thus the energy exchange is slightly in the birds favour.

8: The bird can maintain height in a uniform horizontal wind although the wind has lost speed after the bird has passed.

9: Dynamic soaring probably works better over the oceans than over the land because the wind gradient is smaller and the wind is stronger and steadier

10: The Windward Turn Theory generates a mathematical model which in turn produces a representation of albatross flight which is similar to that seen in nature. It explains how they are able to maintain average airspeed and height over long distances. It helps to explain some aspects of albatross and petrel physiology such as tube nostrils and high aspect ratio wings and of behaviour such as the direction of large scale flight patterns and the shape of small scale repetitive turns. The relatively low G forces help to explain the albatross’ low in-flight metabolism

 

Quite simple really!