Induced drag

Schematic representation of lift distribution, flow around the wing tips and tip vortices

A body exposed to the flow of a fluid experiences resistance . This resistance can be broken down into individual components that have various causes.

One of these components is the induced drag (also: induced air drag). The induced drag always arises when an object in a fluid creates forces across the direction of flow. This is the case, for example, with the generation of lift by the wings of an aircraft. The air is accelerated downward ( downwash ) and on the wing caused tip vortex , which compensate for the pressure difference between the top and bottom. The kinetic energy which is then supplied to the air, lost plane. Contrary to what is often assumed, the induced drag in aircraft does not only arise at the wing tips, but also on the entire wing or every surface that generates lift (transverse force to the direction of flow).

The induced resistance is added to the total resistance with the surface resistance (friction) and the form resistance (end face).

Induced drag in an airplane

Flow course around a wing.
The flow undergoes a change of direction before and after the wing.

The induced drag can be illustrated particularly well on the wing of an aircraft . The aircraft generates dynamic lift by moving a wing through the air at a certain angle of attack and thereby deflecting air downwards ( downwash ). Since with this deflection even a smooth interaction with the wing cannot increase the speed of the air, the deflection downwards must reduce the speed of the horizontal component and thus cause a drag force. This is part of the "induced drag". Another part results from pressure equalization as follows. When generating lift, there is a relative negative pressure above the wing and a relative overpressure below the wing . The areas with the different pressure ratios collide at the ends of the wings, and there is a balancing flow from the area of ​​higher pressure to the area of ​​lower pressure, i.e. from the bottom to the top. Two opposing tip vortices arise at the wing tips , which do not contribute to lift. However, the constant generation of these eddies requires energy and is part of the "induced drag".

calculation

With optimal lift distribution (see below), the drag coefficient is only dependent on the wing extension and the lift coefficient according to the formula:

${\displaystyle C_{W_{i}}={\frac {{C_{A}}^{2}}{\pi \Lambda }}}$         ( is the lift coefficient , is the aspect ratio ) ${\displaystyle {C_{A}}}$ ${\displaystyle {\Lambda }}$

The induced drag can also be given in the form of the additional angle of attack compared to the pure profile data (with Λ = ∞):

${\displaystyle \alpha _{\,\mathrm {ind} }={\frac {C_{A}}{\pi \Lambda }}}$     (in rad ) or          (in angular degrees ) ${\displaystyle \alpha _{\,\mathrm {ind} }={\frac {C_{A}\cdot 57{,}3}{\pi \Lambda }}}$

In slow flight aircraft, the induced drag accounts for more than 50% of the total drag.

Elongation

The airflow at the outermost part of the wing is most affected by the equalizing airflow. Wings with a large aspect ratio therefore generate a lower induced drag, inversely proportional to the aspect ratio, than deep wings with a smaller aspect ratio , with the same area and the same lift.

Reduction of the profile lift coefficient

Profile polar are always listed as wings with infinite extension. The induced drag can also be given in the form of a reduction in lift and thus a reduction in power. The formula is:

${\displaystyle c_{a_{\,{\text{gestreckt}}}}=c_{a_{\,{\text{Profilpolare}}}}\cdot {\frac {\Lambda }{\Lambda +2}}}$

Influence of speed

As the speed of an aircraft increases in level flight, the angle of attack becomes smaller, since the lift force always has to compensate for the aircraft weight and thus remains the same. Due to the lower angle of attack, the induced drag is then lower. Conversely, in slow flight, e.g. B. during take-off and landing, the induced drag is greatest.

Lift distribution

Oswald factor

The Supermarine Spitfire had an almost elliptical wing layout

Since the induced drag increases quadratically with the lift coefficient, the lift coefficient should remain as constant as possible over the entire wing. The optimum corresponds to an elliptical lift distribution over the span. The associated optimal wing outline is only roughly elliptical, as the lift is only roughly proportional to the wing depth ( Re number change with the wing depth). This can be derived mathematically from Prandtl's theory of the bearing line (after Ludwig Prandtl ). In this theory, the flow around the wing is modeled as a potential flow, and some assumptions are made that allow an analytical solution to the problem.

As soon as the wing has a different floor plan, for example a rectangular or pointed one, the induced drag increases. The Oswald factor can be viewed as a form efficiency factor and is therefore always less than one. The higher the Oswald factor, the more favorable the geometry of the wing. In the ideal case (ellipse) the Oswald factor is equal to one. Usually it is in the range from 0.6 to 0.9. To optimize the lift distribution, constructive variants such as B. twist , taper , reduced curvature or winglets . Elliptical wing footprints are expensive to manufacture and are found in a few high-performance gliders these days, such as. B. the SZD-55 and SZD-56 . The trapezoidal floor plan represents a good compromise between flight performance and construction costs.

${\displaystyle C_{W_{i}}={\frac {{C_{A}}^{2}}{\pi \Lambda e}}}$      (Oswald factor <1) ${\displaystyle e}$

The Oswald factor is also referred to as the wing efficiency or the wingspan efficiency. In the optimum, this would have the value 1.

k factor

The k-factor was introduced to describe the influence of any lift distribution on the induced drag. With this "form factor" the increase of the induced drag - compared to the optimal elliptical lift distribution - is summarized in one value. In the optimum, this would have the value . ${\displaystyle k={\frac {1}{\pi \Lambda }}}$

${\displaystyle C_{W_{i}}=k\,{C_{A}}^{2}}$

Both factors or approaches are equivalent. The k-factor is defined as follows: ${\displaystyle k={\frac {1}{\pi \Lambda e}}}$

literature

• David Anderson, Scott Eberhardt: Understanding Flight . McGraw-Hill, New York et al. 2001, ISBN 978-0-07-136377-8 .
• Holger Babinsky: How do wings work? In: Gary Williams (Ed.): Physics education . tape 38 , no. 6 . IOP Publishing (United Kingdom), November 2003 ( Online [PDF; 370 kB ; accessed on January 27, 2018]).
• Götsch, Ernst: Aircraft technology . Motorbuchverlag Stuttgart 2003, ISBN 3-613-02006-8
• Hermann Schlichting, Erich Truckenbrodt: Aerodynamics of the aircraft 2 (classic of technology) . Springer Verlag Berlin 2001, ISBN 3-540-67375-X
• Peter Thiede: Aerodynamic drag reduction technologies. Springer, Berlin 2001, ISBN 978-3-540-41911-2 .

Web links

• ABOUT WINGLETS, by Mark D. Maughmer (English, good explanation of the background; PDF; 137 kB)

Individual evidence

1. ^ Phil Croucher: JAR Private Pilot Studies . Electrocution Technical Publishers, 2005, ISBN 0-9681928-2-3 , pp. 2-9 .
2. ↑ Anderson, Eberhardt: Understanding Flight , p. 46
3. ↑ How do wings work? , P. 502
4. ↑ Anderson, Eberhardt: Understanding Flight , pp. 24ff
5. ^ Chris Waltham: Flight without Bernoulli . In: American Association of Physics Teachers (Ed.): The Physics Teacher . No. 36 . AIP Publishing, November 1998, pp. 458 : “The induced drag D arises from the deflection of air downwards; since even a frictionless interaction with the wing cannot increase the speed of the air, then a deflection down must reduce the horizontal component of its velocity and thus cause a drag force. "
6. ↑ EWD, longitudinal stability ... Part 1, replacement image for a wing, formula 1a
7. ↑ Ethirajan Rathakrishnan: Theoretical Aerodynamics . John Wiley & Sons, 2013, ISBN 978-1-118-47937-7 , pp. 345 .
8. ↑ https://tu-dresden.de/ing/maschinenwesen/ilr/ressourcen/daten/tfd/studium/daten/Aerodynamik_Potentialverbindungen.pdf?lang=de
9. ↑ Dr.-Ing. Wolfgang Heinze: Design of Transport Aircrafts I . Ed .: Institute for Aircraft Construction and Lightweight Construction at the TU Braunschweig. Braunschweig August 2012, p. 128 .