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 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 while the average track has a large downwind component.

Figure 1

dynamic soaring perspective

The Windward Turn

 In the windward turn, the albatross maintains height and loses momentum due to the unbalanced drag force. 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 increasing headwind components, 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-speed is a large proportion of the bird’s airspeed and the amount of turn is limited to less than about 90 degrees. During the windward turn there is a large change of wind-angle and of wind-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 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 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 reduces.

wind components 3

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 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. 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 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, 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. In other words, while inertia resists changes to ground-speed, the changing wind components affect the airspeed.

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, 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.

 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. If the wind is constant the angle of drift will increase and the ground-velocity will change in both speed and direction. The change of ground-velocity is caused by components of the unbalanced drag force.

Effect of drag

 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 has to be caused by a centripetal force.

 Force F1 is the drag force causing acceleration of the airspeed. 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 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) but that the tangential components in each frame are different from each other and the centripetal components in each frame are also different from each other

 (Force F1 reduces as the airspeed reduces but for simplicity here can be regarded as constant)

Force components

 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. 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. 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 force F3 which maintains its orientation relative to the bird as it turns. Remember that lift and drag forces are components of the total aerodynamic force in straight flight when the wings are level.

 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.

 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

Effect of changing wind angle

Airspeed and ground-speed components

 See figure 4b. 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 of headwind component H and the rate of decrease of ground-velocity component K..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 ground-speed component dK/dt.

 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)

Effect of changing wind angle 2

 Comparing Figures 3 and 4b, it can be clearly seen that, in straight flight when wind-angle y is constant, then component H is constant and is 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.

 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 gaining height, it 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 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 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 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?

7168365301_1e3e38ddc5  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 tail/headwind-component, which depends upon the rate of change of tail/headwind-component relative to wind-angle. The crosswind position is where the rate of change of head/tail wind with respect to the wind-angle, is at a maximum.  As the turn procedes 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. When the wing is within about half a wingspan of the surface the vertical momentum given to the air is the same, whilst 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 the turn, increases the distance flown and provides the incentive 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. 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 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 changes with the changing wind-angle and the 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, relative to the ground velocity there is now a large propulsive component Ft acting in the direction of the ground-velocity and it is this which increases the birds ground-speed. Component Fc causes the curved path relative to the ground. Component Ft  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 is the same in both the windward and leeward turns, then the average ground track is straight

Leeward turn 4

 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 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 bird’s total drag losses. 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

Leeward turn 5

 In this case, the ground-speed G is increasing under the effect of the propulsive force Ft 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, it is possible to lose a small amount of airspeed in the leeward turn and make up the loss in the windward turn.

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 airspeed to tend to increase both during the climb upwind and the 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 exactly upwind or down wind heading.

Airspeed Sense

6845970664_c1906db5e6 The need to maintain constant or slightly increasing 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 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 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 so that pitot pressure is not exactly the same as 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. 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 added vectorially to the drag force, creates the horizontal resultant force. The angle of bank therefore gives horizontal momentum to the air in the opposite direction to the horizontal resultant force.

Transfer of momentum

 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  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 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 momentum in both turns and for both the bird and the air 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 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 change of momentum is the same, the air loses more speed and 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 birds drag losses. In effect wind speed has been converted into air turbulence in the wake of the bird. The horizontal momentum taken from or 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.

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.

However, while the ground can be treated as an approximate IFR for the puposes of these calculations, 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

Quite simple really!

 

 

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