Dynamic gliding

An airplane flies on a (locally constant) inclined circular path in a wind field and can thus increase its energy. Energy gain as a result of the wind shear arises on those sections of the flight path where the wind component increases in the direction of flow −u, and that means in a formula . ${\displaystyle -{\dot {\mathbf {w} }}\cdot \mathbf {u} >0}$

Dynamic gliding is a flight technique in which a bird or an airplane can gain energy from the wind shear , i.e. a non-constant wind field . Here the bird or the pilot of the aircraft tries to choose the flight path in such a way that the energy gain (due to the wind shear) exceeds the energy loss (due to the flow resistance ) on average over time.

Using dynamic gliding , albatrosses can fly very long distances a few meters above the sea. Dynamic gliding is also practiced by model glider pilots. While the Albatros uses the energy gain for locomotion, in dynamic gliding with glider models the energy gain is used to fly faster and faster on a locally approximately constant orbit.

Working principle

A simple model that can be used to explain the functional principle of dynamic gliding consists of two layers of air with different wind speeds: the aircraft flies in it cyclically from one layer to the other and repeatedly pierces the separating layer between the layers of air. This two-layer model is described in more detail below, in brief: When the separating layer is pierced, the speed above the ground remains constant because of the conservation of momentum, but the speed changes in relation to the air . During the turns, the speed in the air remains constant, while the basic speed changes its sign. In the ideal case without friction, the aircraft would gain twice the speed difference between the two layers of air with each cycle. In general, however, the gain in kinetic energy (or height) will be significantly lower due to friction losses .

Two-layer model

The energy gain of the aircraft can be illustrated using a two-layer model.

Model assumptions

2-layer model: An aircraft flies through two layers with different wind speeds.

• Notation: v is the amount of the speed vector above ground, w is the amount of the wind speed vector.
• There is no wind below a separation layer, i.e. H. w = 0. Above the layer there is wind w to the right.
• The aircraft create no drag while flying either over or under (but not in) the interface.
• When flying through the separation layer, the air forces remain limited regardless of the separation layer thickness.
• The aircraft flies through the separating layer as shown at a very small angle.

Two different state changes

The aircraft experiences two different state changes:

1. Flying either only within the upper layer or only within the lower layer (without touching the separating layer) on any flight path: While the aircraft is within one of these layers, it has (since there is no friction) a total energy that is constant over time is composed of the potential and the kinetic energy (compared to air), namely E / m = 0.5 u u + g h . This means that at two different points that are at the same altitude z , the aircraft always has the same speed versus. Has air - regardless of its direction of flight. The fact that E / m = 0.5 u u + g h is a conservation quantity for flying within a layer can also be seen from the fact that the right-hand side of the last equation in the section “Description with energy balance” for flying in a layer with constant Wind (as given here) is zero.
2. Piercing the separating layer: When flying through the separating layer, the momentum remains, i.e. the speed compared to Reason received. This is due to the fact that the (resistance) force that acts on the aircraft when it flies through the layer is finite and that at the same time the time it takes to fly through the layer is very short, see Impact of Force . The layer can finally be made thinner and thinner (and / or the aircraft faster and faster) until the momentum of the aircraft does not change at all when flying through, so that v remains constant.

Description of the states in four places

• Position 1: Fly into the separating layer: speed vs. Reason is (initial speed). Since there is no wind, the speed is compared to Air . ${\displaystyle v=v_{1}}$ ${\displaystyle u=v=v_{1}}$
• Digit 2: The speed vs. Air has risen , ground speed continues (airplane flies to the left and has a headwind). ${\displaystyle u=v+w}$ ${\displaystyle v=v_{1}}$
• Position 3: Here is and (plane flies to the right and has a tailwind, before it flew to the left ). ${\displaystyle u=v_{1}+w}$ ${\displaystyle v=v_{1}+2w}$ ${\displaystyle v=v_{1}}$
• Digit 4: Now is and (regardless of the flight direction). ${\displaystyle u=v_{1}+2w}$ ${\displaystyle v=v_{1}+2w}$

This process can now be repeated. The kinetic energy of the aircraft continues to increase. Depending on whether one considers the kinetic energy in an inertial system (for example "above ground") or in relation to the local air (see next section), the increase in energy takes place in the (upper) bend (elastic collision) or in each case sudden increase in the airstream when passing into the other air layer.

Energy balance

The energy gain of the aircraft can also be illustrated by evaluating the pulse rate.

Reference system, speeds and forces

The reference system is free of apparent forces except for the acceleration due to gravity . It can be a coordinate system connected to the ground ("above ground") or one that moves constantly with the mean speed of the wind.

In such a reference system, let the speed vector of the aircraft and the speed vector of the air at the location where the aircraft is currently located, so that the airspeed is. The time derivative of the speed , which is essential for the effect , contains both the actual change in the wind field over time and the mostly dominant contribution of the directional derivative . ${\displaystyle \mathbf {v} }$ ${\displaystyle \mathbf {w} }$ ${\displaystyle \mathbf {u} =\mathbf {v} -\mathbf {w} }$ ${\displaystyle \mathbf {w} }$ ${\displaystyle {\dot {\mathbf {w} }}}$

Lift and drag are used as specific quantities (unit N / kg), i.e. as accelerations like the acceleration due to gravity , so that the mass of the aircraft can be reduced. ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {W} }$ ${\displaystyle \mathbf {g} }$

The drag is by definition parallel to the flow and the lift perpendicular to it. The latter can be expressed using the scalar product : ${\displaystyle \mathbf {-} u}$

${\displaystyle \mathbf {A} \cdot \mathbf {u} =0\,.}$

Impulse and energy balance

According to the law of momentum applies

${\displaystyle m{\dot {\mathbf {v} }}=m\mathbf {A} +m\mathbf {W} +m\mathbf {g} \,,}$

so

${\displaystyle {\dot {\mathbf {v} }}=\mathbf {A} +\mathbf {W} +\mathbf {g} \,.}$

To obtain an energy balance , the last equation is scalar multiplied by and inserted. It follows ${\displaystyle \mathbf {u} }$ ${\displaystyle \mathbf {A} \cdot \mathbf {u} =0\,}$

${\displaystyle {\dot {\mathbf {v} }}\cdot \mathbf {u} =\mathbf {W} \cdot \mathbf {u} +\mathbf {g} \cdot \mathbf {u} }$

Inserting v = w + u yields

${\displaystyle ({\dot {\mathbf {w} }}+{\dot {\mathbf {u} }})\cdot \mathbf {u} \mathbf {\,} =\,\mathbf {W} \cdot \mathbf {u} +\mathbf {g} \cdot (\mathbf {v} -\mathbf {w} )}$

If the amount of with g referred to and is written parallel component of velocity as the rate of change of height, so is thus obtained ${\displaystyle \mathbf {g} }$ ${\displaystyle \mathbf {g} }$ ${\displaystyle \mathbf {g} \cdot \mathbf {v} =-g{\dot {h}}\,.}$

${\displaystyle {\dot {\mathbf {u} }}\cdot \mathbf {u} +g{\dot {h}}=\mathbf {W} \cdot \mathbf {u} -\mathbf {g} \cdot \mathbf {w} -{\dot {\mathbf {w} }}\cdot \mathbf {u} }$

Application of the product rule results in a balance of the 'specific energy compared to air'

${\displaystyle {\frac {d}{dt}}\left({\frac {\mathbf {u} \cdot \mathbf {u} }{2}}+gh\right)\quad ={\underbrace {\,\mathbf {W} \cdot \mathbf {u} \,} _{\text{Widerstand}}}-{\underbrace {\,\mathbf {g} \cdot \mathbf {w} \,} _{\text{Aufwind/Abwind}}}-{\underbrace {\,{\dot {\mathbf {w} }}\cdot \mathbf {u} \,} _{\text{Windscherung}}}}$

The representation is also useful

${\displaystyle {\frac {d}{dt}}\left({\frac {\mathbf {v} \cdot \mathbf {v} }{2}}+gh\right)\quad =\mathbf {W} \cdot \mathbf {u} +(\mathbf {A+W} )\cdot \mathbf {w} =\mathbf {W} \cdot \mathbf {u} -\mathbf {g} \cdot \mathbf {w} +{\dot {\mathbf {v} }}\cdot \mathbf {w} }$

Evaluation of the result

The 'specific energy compared to air' is the sum of 'kinetic energy compared to air' and potential energy. This energy is important for gliders (or birds). An excess of 'kinetic energy compared to air' (too high speed compared to air) can be converted into altitude and vice versa. For this reason, variometers also show the change in precisely this energy (except for the factor mg).

The change in the 'specific energy compared to air' over time is influenced by the three terms on the right: The first term is the specific power due to the resistance. This term is always negative. The middle summand is the specific power from updraft or downdraft. When there is an updraft . That is, updrafts increase the aircraft's energy. The right term is the specific power as a result of the wind shear. It shows that energy gain from wind shear arises on those sections of the flight path where the wind component increases in the direction of flow −u, i.e. H. there where . ${\displaystyle -\mathbf {g} \cdot \mathbf {w} >0}$ ${\displaystyle -{\dot {\mathbf {w} }}\cdot \mathbf {u} >0}$

The last equation shows: If the vector of the total air force (acceleration) points in the direction of w (the wind speed compared to the mean wind), this increases the energy 1/2 v · v + gh . To avoid misinterpretation, v and w should be defined as speeds relative to the mean wind speed.

Appropriate trajectories

Not every trajectory can generate energy from a wind field. But with a given wind field and aircraft, trajectories can be compared to determine how large the energy gain (per time) is.

If there is a horizontal wind that is constant in direction but increases with height, as shown in the picture above, a suitable flight path is e.g. B. an inclined circle whose lowest point is also the furthest downwind. The upper semicircle can be 'folded down' so that an albatross or an airplane can move across the wind.

Even if the wind blows vertically and is not constant (such as in a updraft field ), the aircraft can generate energy from this shear. For example, if the updraft field is strong inside and becomes weaker towards the edge, it is more favorable to fly 'down' into the updraft field and fly out of it 'upwards' than the other way round.

literature

• Thomas, Fred: Fundamentals of Sailplane Design . College Park Press, 1999, ISBN 978-0-9669553-0-9 ( limited preview in Google Book Search).

Web links

• rcspeeds.com model airplane website

Individual evidence

1. ↑ Lord Rayleigh : The Soaring of Birds . In: Nature . tape 27 , 1883, p. 534-535 , doi : 10.1038 / 027534a0 ( nature.com [PDF; accessed December 2, 2014]).
2. ↑ JA Wilson: Sweeping flight and soaring by albatrosses . In: Nature . tape 257 , 1975, pp. 307-308 , doi : 10.1038 / 257307a0 ( nature.com [accessed December 2, 2014]).
3. ^ H. Weimerskirch, T. Guionnet, J. Martin, SA Shaffer, DP Costa: Fast and fuel efficient? Optimal use of wind by flying albatrosses . In: The Royal Society Proceedings B . tape 267 , no. 1455 , 2000, doi : 10.1098 / rspb.2000.1223 ( royalsocietypublishing.org [accessed December 2, 2014]).
4. ^ Gottfried Sachs: Minimum shear wind strength required for dynamic soaring of albatrosses . In: IBIS - international journal of avian science . tape 147 , 2005, pp. 1–10 , doi : 10.1111 / j.1474-919x.2004.00295.x .
5. ↑ Alexander Knoll: Investigation into the manned dynamic glider flight . Herbert Utz Verlag Wissenschaft, 1995, ISBN 3-931327-95-7 ( limited preview in the Google book search).
6. ↑ ingo Martin: Dynamic glider flight for small aircraft. (PDF, 5.6 MB) Student thesis. Institute for Flight System Dynamics, RWTH Aachen University, April 11, 2011, accessed on December 8, 2014 . 
7. ↑ Dynamic gliding. RC-Network.de, accessed December 8, 2014 .