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 windward turn starts with a tailwind component at point A (figure 1) and ends with a headwind component at point C. There is a reversal of the direction of turn and the leeward turn starts with a headwind at point C and ends with a tailwind at point E. The direction of turn reverses again before the next windward turn.
The Windward Turn
In the windward turn, the albatross maintains height and loses momentum due to the unbalanced drag force, expending kinetic energy rather than potential energy. The loss of momentum is seen as a loss of ground-speed rather than a loss of airspeed. Airspeed is constant because the tendency to lose airspeed due to drag is balanced by the tendency to gain airspeed from the reducing tailwind/increasing headwind components, whilst turning relative to the wind.
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-speed is a large proportion of the birdís airspeed and the amount of turn is limited to less than about 90 degrees. During the windward turn there is a rapid and large change of wind-angle and of wind-component from tailwind to headwind. The triangle of velocities does not of itself change the velocities. Those changes are caused by applied forces like lift, drag or gravity.
The bird maintains height and loses momentum due to drag. 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 and the birdís airspeed is 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 is changing in both speed and direction. The change in direction of the birdís air-velocity has the same effect on the birdís airspeed as an opposite change in the direction of the wind-velocity.
How can the bird, in level flight, maintain itís airspeed? Quite simply, you can see in figure 2 that if the wind-angle changes at the correct rate then, the ground-velocity can change direction and the ground-speed can reduce due to the drag force, while the airspeed will be constant. This requires a little more explanation.
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.
In turbulence, acceleration of the air results in corresponding changes in air-velocity while changes of ground-velocity (acceleration of the aircraft) are resisted by inertia.
When the wind is changing from one state of equilibrium to another, the air is accelerated and the 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 airspeed has returned to its original condition. For example, if the headwind increases, airspeed increases slightly, drag increases, ground-speed and airspeed reduce and a new state of equilibrium is reached with the original airspeed but with reduced ground-speed.
In other words, while inertia resists changes to ground-speed, the changing wind components affect the airspeed. Those changing wind-components can be due to changing wind-velocity or changing aircraft-velocity relative to the wind, eg. changing wind-angle.
We also know that if an aircraft is accelerated into an increasing headwind, for example a winch-launched glider, 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 causes the air-speed to reduce and the effect of the wind is to make the ground-velocity change in both speed and direction.
The bird is in a straight glide relative to the air but as the airspeed reduces, the drift angle increases and the path over the ground turns to the right. The bird is not turning relative to the air but itís path over the ground is a curve. That curve has to be caused by a centripetal force.
Force F1 is the drag force. Force components Ft and Fc correspond to tangential and centripetal accelerations of the ground-velocity. Ft is less than F1 and therefore 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 constant wind causes different tangential and centripetal components of acceleration in air and ground frames of reference.
(Force F1 reduces as the airspeed reduces)
The effect of changing wind-angle
In dynamic soaring the bird is never in a state of equilibrium, it is always turning, accelerating and decelerating and of course, there is no thrust. In figure 4a, the bird is turning relative to the wind and maintaining height. Force F1 is drag and force F2 is the centripetal force - the horizontal component of lift due to the angle of bank. The vertical lift force is not shown.
F1 and F2 are combined to create force F3 which maintains its orientation relative to the bird as it turns. Remember that forces F1 (drag) and the lift force only exist as components of the total aerodynamic force in straight flight when the wings are level. In a banked turn the the lift force has a horizontal component which, combined with the drag force, effectively changes the direction of the horizontal acceleration.
At Position 2, an increment of time later, the wind-angle and ground speed have reduced. There is a different solution to the triangle of velocities, in which the airspeed and wind-speed are the same as at Position 1.
How is airspeed constant?
See Figure 4b. Force F3 can now be divided into new components: tangential force Ft and centripetal force Fc, oriented to the direction of the ground acceleration. The ground-speed reduces due to the tangential component force Ft and the ground direction changes due to force Fc
Under acceleration, the airspeed comprises, in effect, a ground speed component VT and a wind speed component H. The tendency for the airspeed to reduce due to drag force F1 is balanced by the tendency to increase due to the changing wind-angle y. In effect, the acceleration of airspeed is the sum of the rate of increase of headwind component H and the rate of decrease of ground-velocity component VT.
The rate of change of airspeed is zero because the rate of increase of the headwind component dH/dt equals the rate of reduction of the component of ground-velocity dVT/dt. The component-of-ground-velocity VT is reducing under the effect of force F1. The increase of headwind component is due the rate of change of the wind-angle. For every wind-angle there will be a rate of turn that matches the rate of increase of headwind component to the rate of reduction of ground-speed and keeps the airspeed constant.
There is no direct link between these two parameters other than the birdís selected rate of turn which is achieved by flying at an angle of bank appropriate to its airspeed.
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, it turns toward the wind by increasing its rate of turn. If airspeed increases, it reduces its angle of bank and reduces its rate of turn.
As the albatross comes around into wind, initially at best glide speed (Figure 2 point P4), the bird 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.
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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 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 gain of potential energy from KE is greater 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 itís 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?
The end of the windward turn comes when the birdís ability to maintain height and airspeed diminishes. This is because the rate of change of headwind component reduces, while drag and the rate of change of ground-speed is approximately constant. (Figure 2 points P3 & P4). Maintaining airspeed depends upon the ability to match the rate of loss of ground-speed with the rate of change of tail/headwind-component , which depends upon the rate of change of tail/headwind-component relative to wind-angle.
The cross-wind position (Figure 2 points P2 & P3) is where the rate of change of head/tail wind with respect to the wind-angle, is at a maximum. As the bird turns toward the wind, the wind-angle reduces to less than about 50deg and the rate of change of wind-component relative to wind-angle rapidly reduces.
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.
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 wingís down-wash. 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 improves the efficiency of the turn without actually adding energy.
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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. It 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 component of lift and the drag force.
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. The wing-over gives the bird time to accelerate, to keep up with the wind as it turns downwind.
(A simple flat downwind turn, at constant airspeed, will cause a loss of height. This is because the acceleration of ground speed, whilst turning downwind requires a force and the only force then available would be gravity).
Figure 5 is a plan view of the leeward turn with the wind coming from the top. It can be seen that when the wind is a large proportion of the airspeed, the drift angle (the angle between the aircraft heading and its actual track across the ground) is also large. Notice how the shape of the triangle changes with the changing wind-angle and the acceleration of ground velocity.
The horizontal, centripetal component of lift combined with the drag force creates the horizontal resultant. 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. Airspeed losses, particularly during the turn reversals, will be offset by the effect of the wind-gradient.
Here is the key point: because of the large drift angle there is also a large propulsive component of the horizontal resultant acting in the direction of the ground-velocity and it is this which increases the birdís ground-speed. This component 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 KE without losing height and it reverses the direction of turn into the next windward turn.
When the bird is gaining height, the propulsive force has a vertical component and the bird will gain extra height and potential energy. The total gain of kinetic and potential energy is equivalent to the birds total drag losses.
How is airspeed the same at the beginning and end of the leeward turn?
In the leeward turn the airspeed effectively comprises a ground speed component and a wind component. When the acceleration of the ground-speed component is equal and opposite to the acceleration of the wind component, the acceleration of airspeed is zero. see figure 6
In this case the ground-speed component is increasing under the effect of the propulsive force while the headwind component reduces under the effect of the increasing wind-angle. The effect of drag (combined with force Lh) is to change the ground velocity. No force is necessary to account for constant airspeed which is achieved by a balance of accelerations.
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 airspeed to tend to increase during the climb upwind and descent downwind which will offset any net loss of airspeed due to drag. It will not necessarily increase the airspeed. It will only be a small effect because the bird is never near to an upwind or down wind heading.
Airspeed Sense .
The need to maintain constant airspeed in the windward turn and the need to sense when airspeed is changing, is critical and controls the birdís management of the manoeuvre. The points of transition from the windward to leeward turns 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 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. The picture shows a giant petrel and itís 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 bird velocity and kinetic energy is proportional to velocity squared. See figure 7
In flight, a wing gives the air downward and forward momentum. The equal and opposite reaction is a force on the wing whose upward component is called lift and whose horizontal component, opposite to the direction of flight, is called drag. When an aircraft banks, the lift force provides an additional horizontal component which provides the centripetal force which makes the aircraft turn.
That horizontal, centripetal component of lift Lh (not the drag force) gives horizontal momentum to the air opposite to the centripetal direction.
In the leeward turn a component of that momentum is parallel to and opposite to the wind direction, therefore the momentum of the wind is reduced whilst the horizontal and vertical momentum of the bird is increased.
During the windward turn, the opposite effect occurs and the horizontal momentum given to the wind 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 momentum in both cases is measured relative to the same ground frame of reference. The windward turn is flown with a small angle of bank and a small horizontal component of lift and therefore the exchange of momentum is at a slow rate.
The bird maintains average speed and height during successive turns and 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 the air loses more energy in the leeward turn than it gets back in the windward turn because, in relation to velocity, kinetic energy is a square law and momentum is a direct law.
The difference in energy is equivalent to the birdís drag losses. In effect wind speed has been converted into air turbulence in the wake of the bird.
The horizontal momentum given to the air affects only the air with which the bird is in contact. 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.
1. In a Windward Turn, the albatross can maintain airspeed because the tendency to lose airspeed due to drag is balanced by the tendency to increase airspeed caused by the changing tail/head-wind component. These changing tail/headwind components are due to the bird turning relative to the wind. The loss of ground-momentum is caused by the unbalanced drag force due to maintaining height.
2. The bird can maintain height because, rather than losing potential energy due to losing height, instead it loses an equivalent amount of kinetic energy due to deceleration of ground-speed.
3. The efficiency of the windward turn is improved by flying in ground-effect.
4. In a leeward turn, aerodynamic forces combined with a large angle of bank and a large angle of drift provide a downwind acceleration enabling acquisition of ground KE without gaining airspeed or losing PE. During the climb, extra potential energy is gained due to the propulsive force.
5. The efficiency of the leeward turn is improved by the wind gradient but does not depend upon it.
5. Wind energy is transferred from the wind to the bird to balance drag losses, by transfer of momentum. Provided that the bird gains more energy in the leeward turn than it loses in the windward turn, then it can maintain or gain height.
6. The Windward Turn Theory helps to explain some aspects of bird physiology such as tube nostrils and high aspect ratio wings and of behaviour such as the direction of large scale flight patterns the shape of the dynamic soaring manoeuvre and flight close to the surface.
Quite simple really!
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Click here for the next page which is a mathematical analysis of the manoeuvre Good luck with that!